GIFT   OF 


PHYSICAL   MEASUREMENTS 


MINOR 


PART  I.    HEAT,  MECHANICS  AND 

PROPERTIES  OF  MATTER 

1916-1917 


PHYSICAL   MEASUREMENTS 


WETZEL  BROS.  PRINTING  CO. 

2110  ADDISON  STREET 

BERKELEY,  CAL. 


PHYSICAL  MEASUREMENTS 

A  LABORATORY  MANUAL  IN  GENERAL  PHYSICS 
FOR  COLLEGES 

BY 
RALPH  S.  MINOR,  Ph.  D. 

Associate  Professor  of  Physics,   University  of  California 


PART  I. 
HEAT,  MECHANICS  and  PROPERTIES  of  MATTER 


In  Collaboration  With 
Wendell  P.  Roop,  A.  B. 

and 
Lloyd  T.  Jones,  Ph.  D. 

Instructors  in  Physics,   University  of  California 

BERKELEY,  CALIFORNIA 
191.6 


GO  3  7 


Copyrighted  in  the  year  1916  by 
Raloh  S.  Minor 


PREFACE 

This  manual  presents  the  laboratory  exercises  of  the 
Courses  in  General  Physics  offered  at  the  University  of 
California.  It  is  printed  in  four  parts,  each  part  accom- 
panied by  appropriate  tables.  Parts  I  and  II  contain 
experiments  on  Heat,  Properties  of  Matter,  and  Mechan- 
ics. Part  III  deals  with  Magnetism  and  Electricity,  and 
Part  IV  is  devoted  to  Sound  and  Light. 

While  the  manual  is  largely  based  upon  the  exercises 
printed  in  Physical  Measurements  in  Properties  of  Matter 
and  Heat-Minor  and  Elston,  and  Physical  Measurements 
in  Sound,  Light  and  Electricity-Minor,  it  represents  a 
revision  in  which  the  writer  has  attempted  two  things. 
First,  to  base  the  college  laboratory  work  squarely  upon 
the  foundation  afforded  by  a  first  course,  and  second,  to 
omit  exercises  based  upon  topics  which  properly  belong 
to  the  lecture  demonstration  and  which  experience  has 
shown  to  be  adequately  treated  there. 

The  writer's  predecessors,  Professor  Harold  Whiting, 
Professor  E.  R.  Drew,  Professor  Elmer  E.  Hall,  Dr.  T. 
Sidney  Elston,  A.  C.  Alexander,  G.  K.  Burgess,  Bruce 
V.  Hill,  A.  S.  King,  and  T.  C.  McKay  have  all  influenced 
the  development  of  these  exercises. 

Material  has  been  freely  drawn  from  the  literature  of 
the  subject  whether  in  magazine  or  text  book  form  with- 
out specific  credit  being  always  indicated. 

RALPH  S.  MINOR 
University  of  California, 
August,  1916. 


3G2703 


LIST  OF  EXPERIMENTS 

1.  Sensitive  Beam  Balance    -  -  12 

2.  Density  of  Air  -  16 

3.  Relative  Density  of  Carbon  Dioxide  -  19 

4.  Volumenometer  -  20 

5.  Surface  Tension  by  Jolly's  Balance  -  23 

6.  Capillarity.     Rise  of  Liquids  in  Tubes  -  25 

7.  Viscosity.     Flow  of  Liquids  in  Tubes  -  28 

8.  Young's  Modulus  by  Stretching  -  30 

9.  Comparison  of  Alcohol  and  Water  Thermometers  33 

10.  Coefficient  of  Expansion  of  a  Liquid  by  Reg- 

nault's  Absolute  Method  -     36 

11.  Coefficient  of  Expansion  of  a  Liquid  by  Pycnom- 

eter  Method  -     38 

Plotting  of  Curves  -     41 

12.  Coefficient  of  Expansion  of  Glass  by  Weight- 

Thermometer  Method     -  -  42 

Calorimetry  -  44 

13.  Specific  Heat  of  a  Liquid  by  Method  of  Heating  46 

14.  Specific  Heat  of  a  Liquid  by  Method  of  Cooling  48 

15.  Heat  of  Fusion  of  Wood's  Metal  -  50 

16.  Fuel-Value  of  Alcohol  -  53 

17.  The  Principle  of  Moments:  Statics  55 

18.  The  Parallelogram  Law:  Statics  -  58 

REFERENCE  BOOKS 

Duff:  Text-Book  of  Physics  (Fourth  Edition). 

Edser :   Heat  for  Advanced  Students. 

Gariot:  Text-Book  of  Physics  (18th  Edition). 

Kimball:  College  Physics. 

Lodge:  Elementary  Mechanics. 

Preston:  Theory  of  Heat  (Second  Edition). 

Watson:  Text-Book  of  Practical  Physics. 

Kaye  and  Laby:  Physical  and  Chemical  Tables. 

Landolt  and  Bernstein:  Physical  and  Chemical  Tables. 

Smithsonian  Institute:  Physical  Tables. 


INTRODUCTION 

This  book  is  intended  to  be  mainly  a  manual  of  direc- 
tions. It  is  complete  enough,  however,  so  that  when  used 
in  conjunction  with  Duff's  Physics,  the  text-book,  it 
will  cover  the  minimum  requirement  for  the  year's  work. 

Experiments  are  assigned  on  the  bulletin  boards.  Note 
your  assignment  and  find  out  what  you  are  going  to  do 
before  you  enter  the  laboratory.  The  necessary  apparatus 
may  be  obtained  from  the  store-keeper.  In  writing  re- 
ports, use  the  paper  and  binders  sold  for  the  purpose. 
Take  the  data  in  duplicate,  and  drop  the  carbon  copy  in 
the  box  in  the  hall  on  leaving.  Make  your  report  concise. 
Criticize  your  results.  If  they  are  not  satisfactory  to 
you,  explain  why. 

Within  one  week  after  you  have  finished  taking  data, 
drop  your  completed  report  in  the  box  in  the  hall  under 
your  recitation  instructor's  name.  In  writing  the  report, 
follow  the  general  directions  printed  in  the  binder.  All 
curves  must  be  plotted  on  millimeter  paper. 

Your  report  will  be  returned  to  you  by  your  recitation 
instructor.  If  it  is  graded,  note  corrections,  and  keep  it. 
If  it  is  not  graded,  make  indicated  corrections,  and  return 
it  as  soon  as  possible. 

Errors 

Every  measurement  is  subject  to  errors.  In  the  simple 
case  of  measuring  the  distance  between  two  points  by 
means  of  a  meter  rod,  a  number  of  measurements  usually 
give  different  results,  especially  if  the  distance  is  several 
meters  long  and  the  measurements  are  made  to  fractions 
of  a  millimeter.  The  errors  here  arise  from  inaccuracy 


8  SIGNIFICANT  FIGURES 

in  setting,  inaccuracy  in  estimating  the  fraction  of  a 
division,  parallax  in  reading,  faults  in  the  meter  rod,  ex- 
pansion due  to  change  in  temperature,  and  so  on.  Blun- 
ders in  putting  down  a  wrong  reading  or  in  not  adding 
correctly  are  not  classed  as  errors. 

SIGNIFICANT  FIGURES 

It  is  necessary  always,  in  recording  measurements,  to 
put  down  all  the  figures  that  can  be  trusted,  even  if  some 
of  them  are  ciphers.  For  instance,  if  a  distance  has  been 
measured  to  hundredths  of  a  centimeter  and  found  to  be 
50.00  cm.,  it  is  not  correct  to  put  the  distance  down  as 
50  cm.,  for  the  statement  that  the  distance  is  50  cm.  implies 
that  the  distance  has  been  only  roughly  measured  and  is 
found  to  be  more  nearly  equal  to  50  cm.  than  it  is  49  or  51 
cm.,  whereas  the  statement  that  the  distance  is  50.00  cm. 
implies  that  the  distance  lies  between  49.99  and  50.01  cm. 
When  the  distance  is  said  to  be  50  cm.,  the  implication  is 
that  the  second  figure,  counting  from  the  left,  is  the  last  in 
which  any  confidence  can  be  placed;  but  when  the  distance 
is  said  to  be  50.00  cm.,  the  implication  is  that  confidence 
can  be  placed  in  the  quantity  as  far  out  as  the  fourth 
figure  from  the  left.  This  fact  is  indicated  by  saying  that 
in  the  first  case  the  quantity  can  be  trusted  to  the  second 
significant  figure,  and  in  the  other  case  to  the  fourth. 

The  number  of  significant  figures  in  a  quantity  is  irre- 
spective of  the  decimal  point;  thus  there  is  one  significant 
figure  in  0.01,  two  in  0.000026,  and  three  in  2.10.  A 
convenient  method  of  writing  a  number  so  as  to  show  its 
precision  is  to  place  the  point  after  the  first  significant 
figure  and  multiply  by  an  appropriate  positive  or  negative 
power  of  ten.  If  a  mass,  for  instance,  is  about  25,000  gm. 
and  the  third  significant  figure  is  the  last  in  which  any 


SIGNIFICANT  FIGURES  9 

confidence  can  be  placed,  this  fact  is  indicated  by  saying 
that  the  mass  is  2.50  X  104  gm.  The  power  of  ten  is  the 
characteristic  of  the  logarithm  of  the  number. 

In  indirect  measurements  the  result  is  usually  calculated 
by  some  formula.  To  find  how  many  figures  should  be 
kept  in  the  result  and  how  accurately  the  several  quantities 
in  the  formula  should  be  measured,  the  student  should 
observe  the  following  rules. 

Sums  and  Differences. 

In  sums  and  differences  no  more  decimal  places  should  be 
retained  than  can  be  trusted  in  the  quantity  having  fewest 
trustworthy  decimal  places.  Also, 

Quantities  which  are  to  be  added  or  subtracted  should  be 
measured  accurately  to  the  same  decimal  place,  irrespective 
of  whether  they  have  the  same  number  of  significant  figures 
or  not. 

Products  and  Quotients. 

In  products  and  quotients  no  more  significant  figures 
should  be  kept  than  can  be  trusted  in  the  factor  having  fewest 
trustworthy  figures.  Also, 

Quantities  which  are  to  be  multiplied  or  divided  should  be 
measured  to  the  same  number  of  significant  figures. 

Until  the  final  result  is  reached,  it  is  often  worth  while 
to  keep  one  more  figure  than  the  above  rules  indicate. 

In  cases  in  which  a  result  must  be  calculated  from  several 
different  measurements,  it  is  useless  to  take  some  of  these 
measurements  with  high  precision  when  the  precision  of 
the  result  is  limited  by  that  of  some  measurement  whose 
precision  is  necessarily  low.  For  example,  if  you  are 


10  SIGNIFICANT  FIGURES 

measuring  the  area  of  a  rectangle  which  is  ten  times  as 
long  as  it  is  broad,  it  is  a  waste  of  time  to  determine  the 
length  within  a  tenth  of  a  millimeter  if  the  breadth  can 
be  determined  at  the  best  only  within  a  tenth  of  a  milli- 
meter. For  if  this  were  done,  the  length  would  be  measured 
to  one  more  significant  figure  than  the  breadth,  whereas 
the  precision  of  the  calculated  area  would  not  exceed 
that  of  the  breadth. 

Percent  Error 

The  number  of  significant  figures  used  in  expressing 
a  quantity  is  only  a  rough  indication  of  its  precision.  A 
more  exact  statement  of  the  precision  is  made  in  terms  of 
percent  error. 

The  error  in  a  quantity  may  be  estimated  by  comparing 
two  successive  determinations.  The  deviation  of  either 
from  the  mean  indicates  the  order  of  magnitude  of  the 
probable  error.  Or,  if  several  determinations  are  made, 
the  average  deviation  of  the  different  determinations 
from  their  mean  may  be  taken  as  the  probable  error  of 
a  single  observation.  This,  divided  by  the  number  of 
observations,  gives  the  approximate  probable  error  of 
the  mean  value.  The  probable  error  may  be  calculated 
exactly  by  means  of  the  method  of  least  squares,  which 
will  be  taken  up  in  later  work. 

The  error,  as  thus  determined,  is  then  expressed  as  a 
fraction  of  the  quantity  measured,  in  terms  of  percent. 

Do  not  estimate  the  precision  of  your  measurements  by 
comparing  your  calculated  result  with  a  tabular  value. 

Do  not  get  the  impression  that  a  result  apparently  in 
good  agreement  with  a  tabular  value  is  preferable  to  an 


PERCENT  ERROR  11 

honest  reduction  of  data  actually  obtained.  Frequently 
the  tabular  value  would  be  wrong,  by  reason  of  the 
difference  in  the  sample.  But  even  if  you  have  good 
reason  to  believe  that  agreement  with  tabular  results  is 
desirable,  you  will  do  better  to  cultivate  the  manipulative 
skill  which  is  necessary  to  obtain  such  results  rather 
than  to  make  your  data  appear  better  than  they  are. 
Manipulative  skill  is  a  matter  of  experience,  of  which 
you  have  little.  Correctness  of  calculations  is  a  matter 
of  arithmetic,  but  mistakes  are  common.  But  when,  by 
a  "mistake",  poor  data  give  good  results,  it  is  a  matter  of 
fundamental  honesty.  A  clear  scientific  conscience  is 
absolutely  essential  to  the  validity  of  any  sort  of  scien- 
tific work. 

When  the  greatest  errors  involved  are  of  the  order  of 
0.01%,  use  five-place  logarithms;  when  they  are  of  the 
order  of  0.1%,  use  four-place  logarithms.  Careful  work 
with  a  10-inch  slide  rule  will  introduce  errors  not  exceeding 
two  or  three  tenths  of  a  percent. 


12 


WEIGHING  BY  METHOD  OF  VIBRATIONS 


1.     SENSITIVE  BEAM  BALANCE.     DENSITY  OF  A 

SOLID. 

Weighing  by  Method  of  Vibrations. 

When  free  to  swing,  the  pointer  of  an  equal-arm  beam 
balance  oscillates  about  a  mean  position  called  the  rest 
point.  This  is  the  position  it  would  occupy  if  allowed  to 
come  to  rest.  The  rest  point  with  no  load  in  either  pan 
is  called  the  zero  rest  point.  The  construction  of  the  bal- 
ance is  such  that  if  the  two  pans  contain  equal  loads 
the  pointer  will  oscillate  about  this  zero  rest  point,  but 
if  there  is  a  small  excess  in  one  pan,  the  rest  point  will 
be  shifted  slightly  in  the  direction  corresponding  to  a 
lowering  of  the  heavily  loaded  pan. 

To  determine  a  rest  point,  set  the  beam  to  swinging 
freely.  Read  an  odd  number  of  successive  turning  points 
(say  5  or  7),  on  the  scale  behind  the  pointer.  This  scale 
has  zero  on  the  left  and  ten  in  the  center.  The  average 
of  the  mean  of  all  the  left  hand  and  the  mean  of  all  the 
right  hand  readings  is  the  rest  point. 

A  rest  point  determination  by  the  method  of  Vibrations 
not  only  eliminates  the  friction  of  the  knife  edges,  but 


Fig.  1. — The  three  turning  points  shown  in  this  figure 
indicate  13.3  as  the  rest  point. 


1]  METHOD  OF  VIBRATIONS  13 

also  avoids  loss  of  time  which  would  elapse  in  waiting 
for  the  pointer  to  stop. 

In  weighing,  place  the  unknown  mass  in  the  left  pan 
and  load  the  right  pan  with  known  masses  until  a  rest 
point  on  the  scale  can  be  observed.  In  making  trials 
proceed  systematically,  beginning  with  the  large  masses. 

THE  BALANCE  IS  A  DELICATE  INSTRUMENT. 
HANDLE  IT  WITH  CARE.  Keep  the  door  closed,  and 
only  open  it  to  change  masses.  Never  change  masses 
except  when  the  beam  is  clamped.  Handle  the  masses 
with  forceps,  to  prevent  corrosion. 

Use  the  rider  instead  of  masses  of  less  than  10  mg.  The 
rider  weighs  12  mg.,  but  when  placed  at  a  point  on  the 
beam  instead  of  in  the  pan,  is  equivalent  to  a  proportion- 
ately less  amount. 

The  sensitiveness  of  the  balance  is  the  shift  of  the  rest- 
point  per  mg.  excess  in  one  pan.  It  depends  to  a  certain 
extent  on  the  load,  and  must  be  determined  with  the 
pans  loaded.  Having  found  the  sensitiveness,  calculate 
the  amount  which  would  have  to  be  added  to  or  sub- 
tracted from  the  mass  in  the  right  pan  in  order  to  make 
the  rest  point  come  to  its  zero  position. 

An  illustrative  set  of  data  follows. 

From  the  data  in  (ii)  and  (iii)  the  sensitiveness  is  found 
to  be  (14.87-8.54)  /  2  =  3.2  scale-divisions  per  mg.  From 
this  and  the  data  in  (i)  and  (ii)  the  approximate  mass 
of  the  body  is  found  by  calculation  to  be  equal  to  12.2302 
Rm. 

The  reciprocal  of  the  sensitiveness,  called  the  "sensi- 
bility" represents  the  number  of  milligrams  required  to 
shift  the  rest  point  one  division.  From  the  data  given 
in  (ii)  and  (iii)  the  sensibility  is  2/(14.87  -  8.54)  =  0.316 
mg.  per  scale  division. 


14  METHOD  OF  VIBRATION  [1 

(i)     Pans  empty; 

Left  Right 

0.7  15.2 

1.3  14.5 

2.1  13.8 

13.2 


3)4.1  4)56.7 

1.37  14.18 

Zero  rest-point,  7.78 

(ii)     Body  on  left  pan,  12  +  0.200  +  0.020  +  0.008 
12.228  gm.    on  right  pan; 

Left  Right 

10.9  18.6 

11.4  18.2 

11.7  17.8 
12.1 


4)46.1  3)54.6 


11.53  18.20 

Rest-point,  14.87 

(iii)     Body  on  left  pan,  12  +  0.200  +  0.020  +  0.010 
12.230  gm.  on  right  pan; 

Left  Right 

6.3  11.0 

6.5  10.7 

6.7  10.4 

10.2 

3)19.5  4)42.3 


6.50  10.58 

Rest-point,  8.54 


1]  DENSITY  OF  A  SOLID  15 

(a)  Weigh  the  solid,  by  the  method  of  vibrations,  as 
outlined  above.  Record  its  number  and  the  number  of 
the  set  of  weights. 

(6)     Repeat  in  detail. 

(c)  Read  the  barometer  and  the  thermometer. 
Measure  the  dimensions  of  the  cylinder  with  a  metric 
scale  and  calculate  its  volume. 


(d)  From  your  data  in   (a)   and   (b)   find  the   mean 
value  of  the  apparent  mass  of  the  solid. 

The  balance  only  serves  to  compare  forces,  and  hence 
indirectly  masses.  But  the  forces  which  you  have  bal- 
anced against  each  other  in  (a)  are  not  simply  the  weights 
of  the  cylinder  on  one  side  and  of  the  marked  brass 
weights  on  the  other.  The  buoyancy  of  the  air  has  an 
appreciable  influence.  From  the  approximate  volumes 
of  displaced  air  and  the  density  of  air,  (see  page  62), 
calculate  the  bouyant  forces  and  apply  the  correction 
to  the  observed  mass  to  get  the  true  mass. 

(e)  Calculate  the  percent  deviation  of  each  determin- 
ation of  the  mass  from  the  mean  value. 

The  balance  does  not  even  compare  forces,  if  its  arms 
have  not  exactly  equal  lengths,  but  only  force-moments. 
The  error  introduced  by  the  inequality  of  the  arms  may 
be  eliminated  by  re-weighing  with  cylinder  and  masses 
interchanged.  Show  using  moments  that  if  this  be  done, 
the  true  mass  is  given  by 


m  =     /  m'  m" 

S 

where  m'  and  m"  are  the  results  of  the  two  weighings. 
(/)     Calculate  the  density  of  the  solid. 


16  DENSITY  OF  AIR  [2 

2.     DENSITY  OF  AIR. 

To  find  the  density  of  the  air  at  atmospheric  pressure. 

By  the  density  of  a  body  is  meant  the  mass  per  unit 
volume  of  the  body,  and  it  is  usually  found  by  measuring 
the  mass  and  the  volume  and  then  calculating  their 
quotient.  In  the  case  of  air  the  density  may  be  found  by 
an  indirect  method  which  does  not  require  that  the  mass 
of  air  filling  a  given  volume  shall  be  known. 

Let  a  glass  bulb  of  volume  V  be  weighed  full  of  air  at 
atmospheric  pressure  Pi,  and  let  M  be  the  mass  necessary 
to  balance  it.  Then  let  some  of  the  air  be  pumped  out 
until  the  pressure  is  P2,  the  mass  as  determined  by  weigh- 
ing now  being  (M — m),  where  m  is  the  mass  of  air  that 
has  been  pumped  out  between  the  weighings.  Then,  if 
di  and  d2  be  the  densities  corresponding  to  the  pressures 
Pi  and  P2,  it  follows,  from  the  definition  of  density  and 
the  interpretation  of  m,  that 

(1)         Vdl  —  Vd>  =  m. 

The  reciprocal  of  the  density  is  the  volume  of  unit 
mass;  hence,  if  Boyle's  law  is  applied  to  unit  mass  of  the 
air,  the  temperature  being  assumed  constant,  we  have  by 
equating  the  products  of  pressure  by  volume  of  unit  mass 
before  and  after  the  change  in  pressure, 

(2)  Pi/^-P./d,. 
Eliminating  d2  from  (1)  and  (2),  we  get 

(3)  dl 


2]  DENSITY  OF  AIR  17 

In  the  application  of  this  formula  it  is  essential  that  the 
temperature  should  be  the  same  during  the  two  weighings. 
(Why?)  This  condition  is  approximately  satisfied  in 
practice.  If  the  temperature  be  not  the  same,  the  observed 
pressure  in  the  second  case  will  need  to  be  corrected 
(through  the  application  of  Charles'  law),  so  as  to  give 
the  pressure  that  would  have  existed  had  the  temperature 
been  the  same  as  during  the  first  weighing.  The  volume 
V  is  obtained  by  weighing  the  bulb  when  empty  and 
then  when  full  of  water  at  a  known  temperature. 

(a)  Carefully  dry  the  flask  by  exhausting  it  several 
times  and  admitting  air  each  time  through  a  calcium 
chloride  drying-tube.  Ask  an  assistant  for  instructions  in 
regard  to  manipulating  the  pump.  If  moisture  is  visible 
inside  the  flask,  it  may  be  necessary  to  put  in  a  little 
alcohol,  rinse  the  flask,  vaporize  the  alcohol  over  a  Bunsen 
burner,  and  rinse  with  dry  air  as  before.  With  the  dried 
flask  in  connection  with  the  drying  tube,  admit  air  at 
atmospheric  pressure.  Close  the  pinch-cock  and  care- 
fully weigh  the  flask.  Note  the  temperature.  Read  the 
barometer  for  the  pressure. 

(6)  Pump  some  of  the  air  out  until  a  moderately  low 
pressure  is  obtained,  and  weigh  again  at  this  reduced 
pressure.  Again  note  the  temperature  and  record  the 
pressure.  The  pressure  of  the  air  in  the  partially  ex- 
hausted flask  is  equal  to  the  difference  between  the  height 
of  the  barometer  column  and  the  height  of  the  mercury 
column  in  the  manometer  connected  with  the  pump. 

(c)  Repeat  (a)  and  (6)  before  continuing  the  exper- 
iment. 

Fill  the  flask  completely  with  water  up  to  the  pinch- 
cock,  taking  care  to  have  no  water  above  it.  The  temper- 


18  DENSITY  OF  AIR  [2 

ature  of  the  water  should  be  recorded.  Dry  the  outside 
of  the  flask  and  then  weigh  upon  trip  scales  or  a  decigram 
balance. 


(d)  Using  the  data  obtained  in  (a),  (b),  and  (c),  find 
the  density  of  the  air,  in  grams  per  cc.,  at  the  given  tem- 
perature  and   atmospheric   pressure.      From   this   result 
the  density  of  dry  air  under  standard  conditions   (that 
is,  at  0°  C.  and  76  cm.  pressure)  may  be  found  through  the 
application  of  Boyle's  and  Charles'  laws,  or  a  combination 
of  the  l!wo.    If  Pi,  di,  and  7\  represent  the  pressure,  den- 
sity and  absolute  temperature  of  a  given  kind  of  gas  at 
one  time,  and  P2,  d2,  and  T2  represent  the  corresponding 
values  at  another  time,  then  it  follows,  from  a  combina- 
tion of  the  two  laws,  that 

(4)      pl/d1rl  =  p2/j2r2 

for  the  given  kind  of  gas  to  the  degree  of  approximation 
with  which  it  observes  the  given  laws.  Making  use  of 
this  relation,  calculate  the  density  of  dry  air  under  stand- 
ard conditions  of  temperature  and  pressure.  Calculate 
the  probable  error  in  your  result.  Point  out  the  chief 
sources  of  error. 

(e)  Why  is  it  desirable  in  (b)  to  exhaust  the  flask  until 
the  pressure  is  quite  low? 

In  order  that  the  method  of  this  experiment  may  apply, 
is  it  necessary,  or  not,  to  exhaust  the  flask  completely? 
To  what  form  would  the  formula  (3)  be  reduced  in  this 
case? 


3]          RELATIVE  DENSITY  OF  CARBON  DIOXIDE  19 

3.     RELATIVE  DENSITY  OF  CARBON  DIOXIDE. 

The  relative  density  of  carbon  dioxide  compared  with 
air  as  a  standard  is  to  be  measured. 

The  method  employed  here  is  that  used  in  Exp.  2. 
Using  the  same  symbols  as  there  used,  and  making  the 
weighings  and  noting  the  pressures  as  there  indicated, 
we  have  for  the  air, 


(1)  dl 

If  the  measurements  are  then  repeated  for  the  carbon 
dioxide, 

(2)  df  =mfPif/[V(Pif—  P2')], 

the  symbols  having  the  same  meaning  as  in  the  case  of  air. 

From  (1)  and  (2),  if   D  is   the  relative  density  of  the 
carbon  dioxide,  we  get,  by  division, 

(3)         D  =  <*//</, 

=  [m'P/  (P,  —  P,)]  /  [mPi  (P/  —  P/)]  ; 

from  which  we  see  that  a  determination  of  the  volume  of 
the  flask  is  unnecessary. 

(a)  Read  the  directions  given  under  Exp.  2.     Ask  an 
assistant  for  instructions  in  the  use  of  the  pump.     Care- 
fully dry  the  flask,  and  fill  it  with  dry  air  admitted  through 
the  calcium  -chloride  tube.      Using  a  sensitive  balance, 
weigh  the  flask  full  of  air  at  atmospheric  pressure,  noting 
the  pressure  and  temperature.     In  weighing,  follow  the 
method  given  in  Exp.  1. 

(b)  Pump  some  of  the  air  out  until  a  low  pressure  is 
obtained,  and  weigh  the  flask  again  at  the  reduced  pres- 


20  THE  VOLUMENOMETER  [3-4 

sure.  If  the  temperature  is  not    the   same,  within    0°.5, 
the  observed  pressure  should  be  corrected  as  in  Bxp.  2. 

(r)  Fill  the  flask  with  dry  carbon  dioxide  at  atmos- 
pheric pressure.  This  can  best  be  done  by  pumping  out 
the  flask  and  admitting  the  gas  from  the  generator  several 
times  in  succession.  Take  care  not  to  allow  any  air  to 
pass  through  the  acid  into  the  generator;  and  keep  the 
stop-cock  closed  when  not  using  the  generator.  When 
the  flask  is  filled  with  carbon  dioxide  at  a  known  pressure 
and  temperature,  weigh  it  as  before. 

(d)  Pump  some  of  the  carbon  dioxide  out  until  a  low 
pressure  is  obtained,  as  in  the  case  of  the  air,  and  weigh  again. 


(e)  By  the  use  of  equation  (3),  calculate  from  your 
data  the  relative  density  of  carbon  dioxide,  with  respect 
to  air,  under  the  given  conditions  of  temperature  and 
pressure  prevailing  in  the  room. 


4.     THE  VOLUMENOMETER. 

To  find  the  density  of  an  irregular  solid  by  means  of  the 
volumenometer  and  the  balance. 

In  the  volumenometer,  A  is  a  glass  tube  which  may  be 
closed  at  the  top  by  a  ground  glass  plate.  It  corresponds 
to  the  closed  tube  in  the  experiment  on  Boyle's  law.  As 
in  that  experiment,  the  pressure  and  volume  of  air  in  A 
are  varied  by  raising  or  lowering  a  tube  containing  mer- 
cury. The  pressure  is  determined  by  noting  the  differ- 
ence in  the  levels  of  the  mercury  columns  and  sub- 
tracting it  from  the  atmospheric  pressure.  The  volume  is 
unknown.  The  volume  of  a  portion  of  the  tube  between 
the  two  points  M  and  N,  however,  is  known. 


4] 


THE  VOLUMENOMETER 


21 


Let  the  volume  between  M  and  N  be  k,  and  that  above 
M  be  V.  By  determining  the  pressures  when  the  tube  A 
is  full  of  air  at  atmospheric  pressure  and  volume  V 
(mercury  meniscus  at  M),  and  again  when  the  tube  con- 
tains the  same  mass  of  air  expanded  to  occupy  the  volume 
V+k  (mercury  meniscus  at  N),  an  equation  involving 
Boyle's  law  may  be  written  containing  these  two  volumes 


Fig,  2. — The  volumenometer  is  shown,  filled  with  a  certain  mass 
of  air  of  volume  V,  and  then  with  the  same  mass  of  air 
expanded  to  occupy  volume  (V  +  k). 

and  the  corresponding  pressures.  For  instance,  if  Pi  and 
P2  are  the  two  pressures,  we  will  have,  if  the  temperature 
is  constant, 

P,  V  =  P2  (V  +  k). 
From  this  equation  we  get,  by  solution, 

(1)         V  =  P2k/(Pl  —  Pz). 

The  volume  of  the  air-space  in  A  above  M  thus  becomes 
known  in  terms  of  the  two  pressures  and  the  volume  k. 

If  now  the  solid  whose  density  it  is  desired  to  find  be 
placed  in  A,  the  volume  of  the  air-space  in  A  above  M 
will  be  less  than  before  by  an  amount  equal  to  the  volume 


22  THE  VOLUMENOMETER  [4 

of  the  solid.  Let  the  volume  of  the  air-space  now  above 
M  be  V .  By  proceeding  as  before,  we  get 

(2)         V  =  P2'k/(P>'  —  P/), 

where  PI  and  P2'  are  the  two  pressures  in  the  enclosed 
air  corresponding  to  the  volumes  V  and  V'+k,  the  first 
representing  the  atmospheric  pressure.  From  (1)  and 
(2)  the  volume  of  the  solid  can  be  found.  The  density 
of  the  solid  can  then  be  calculated  from  its  volume  and 
mass. 

(a)  With  the  tube  A  uncovered  bring  the  mercury 
meniscus  to  M,   recording  the  pressure,   evidently  just 
equal    to    the    atmospheric    pressure    and  obtained    by 
reading     the    barometer.       Carefully    place    the    plate 
on  A,  so    as    to    insure    an    air-tight   joint.      The    plate 
must  be  clean  and  have  on  it  only  a  little  grease.     By 
lowering  the  open  tube  on  the  side  L  cause  the  mercury 
meniscus  in  the  tube  A  to  drop  from  M  to  N,  then  note 
the  difference  in  mercury  levels  at   TV  and  L  and  again 
determine   the   pressure   in   the   enclosed   air.      Test   for 
leakage  by  allowing  the  tube  to  remain  a  minute  or  more 
in  this  position,  and  make  sure  that  the  mercury  levels 
do  not  change.     Note  the  value  of  k  recorded  on  the  tag 
attached  to  the  apparatus. 

(b)  Remove  the  plate,  place  inside  the  volumenometer 
one  of  the  bodies  whose  density  is  to  be  determined,  and 
repeat  (a).     Weigh  the  body. 

(c)  Repeat  for  at  least  one  other  body. 


(d)     By  applying  Boyle's  law  find  the  volume  of  the 
solids  used  and  then  calculate  their  densities. 

What    are  the  advantages  and  disadvantages  of  this 
method  of  determining  density? 


5]  SURFACE  TENSION  BY  JOLLY'S  BALANCE  23 

5.  SURFACE  TENSION  BY  JOLLY'S  BALANCE. 

To  obtain  a  direct  measure  of  the  surface  tension  of  a 
liquid  by  balancing  it  against  the  tension  in  a  stretched  spring. 

A  wire  rectangle  is  hung  from  the  spring  of  a  Jolly's 
balance  and  allowed  to  dip  in  a  soap  solution  which  forms 
a  film  across  the  rectangle.  When  equilibrium  is  estab- 
lished, the  force  due  to  surface  tension  in  the  two  surfaces 
of  the  film  must  just  balance  the  tension  in  the  spring. 
By  knowing  the  force  which  will  stretch  the  spring  the  same 
amount,  we  have  a  measure  of  the  total  force  due  to  sur- 
face tension.  If  T  is  the  value  of  the  surface  tension  in 
dynes  per  centimeter  width  of  the  film,  /  the  width  in 
centimeters  of  the  rectangle  along  the  surface  of  the 
liquid,  and  F  the  force  in  dynes  exerted  by  the  spring, 
then 

F  -  2 1  r. 

The  force  F  in  dynes  is  equal  to  980  m,  where  m  is  the 
mass  in  grams  whose  weight  will  stretch  the  spring  the 
given  amount  and  980  is  approximately  the  number  of 
dynes  of  force  which  the  earth  exerts  on  1  gm.  Know- 
ing m  and  /,  the  value  of  T  can  be  calculated. 

Bare  wire,  pincers,  thread  and  a  piece  of  emery  cloth 
are  provided  for  the  construction  of  the  rectangles.  The 
greatest  care  must  be  taken  to  have  the  beaker  and 
rectangles  clean.  Do  not  touch  with  the  fingers  the  inside 
of  the  beaker,  the  liquid,  or  the  part  of  the  rectangle  on 
which  the  film  is  formed,  for  a  slight  trace  of  grease  will 
very  greatly  decrease  the  surface  tension  of  water. 

(a)  Having  first  cleaned  the  pincers  and  the  wire 
with  -emery  cloth  construct  three  rectangles  about  2, 
4  and  6  cm.  wide  respectively.  Make  the  rectangles  to 


24  SURFACE  TENSION  BY  JOLLY'S  BALANCE  [5 

approximate  size  and  determine  their  exact  width  after 
you  have  finished  using  them.  These  should  have  the 
form  of  a  staple  with  square,  corners  and  legs  from  3  to  5 
cm.  long.  Suspend  a  rectangle,  2  cm.  wide,  from  the 
spring,  and  let  it  be  partially  immersed  in  a  beaker  of 
soap-solution.  Read  the  extension  of  the  spring  when 
there  is  no  film  in  the  rectangle,  and  again  with  a  film 
across  it. 

Repeat  until  consistent  results  are  obtained,  and 
average. 

Repeat  these  measurements,  using  rectangles  4  cm. 
and  6  cm.  wide. 

Determine  by  trial  whether  the  force  exerted  by  the 
film  depends  on  the  length  of  the  film,  measured  parallel 
to  the  direction  of  the  force. 

(6)  Calibrate  the  balance  by  observing  the  extension 
produced  by  known  weights.  Use  extensions  equal  to 
or  somewhat  larger  than  those  obtained  with  the  film. 

(c)  Make  a  new  rectangle,  4  cm.  wide;  clean  the  beaker 
thoroughly;   and  repeat   (a)   with  water  fresh  from  the 
tap.     As  a  film  of  no  appreciable  height  will  form  with 
water,  take  the  reading  of  the  balance  without  the  film 
when  the  under  side  of  the  upper  wire  of  the  rectangle 
is  just  above  the  surface  of  the  water  and  not  in  contact 
with  it;  and  again,  after  immersing  the  upper  wire  of  the 
rectangle  so  as  to  wet  it,  take  a  reading  when  it  breaks 
away  from  the  surface. 

Repeat  until  consistent  results  are  obtained,  and 
average. 

(d)  Repeat  (c),  using  water  at  50°  C.  or  higher. 

(e)  Repeat  (c),  using  alcohol. 


5-6]  CAPILLARITY  25 

(/")  From  the  data  taken  in  (a),  state  how  the  total 
tension  in  the  film  varies  with  its  width.  Calculate  the 
surface  tension,  T,  in  dynes  per  cm.,  for  the  liquids  used 
in  (a),  (c),  (d),  and  (<?).  Present  your  results  in  tabular 
form. 

6.     CAPILLARITY.    RISE  OF  LIQUIDS  IN  TUBES. 

To  determine  the  surface  tension  of  water  and  of  alcohol 
from  the  rise  of  these  liquids  in  capillary  tubes. 

When  the  inner  surface  of  a  tube  is  wet  by  a  liquid,  the 
surface  tension  of  the  latter  may  be  considered  as  acting 
upward  at  all  points  around  the  circumference  of  the 
tube.  The  total  vertical  component  of  this  force  is 
2wrT  cos  a,  where  r  is  the  radius  of  the  tube  in  cm.,  T  the 
surface  tension  in  dynes  per  cm.,  and  a  is  the  angle  of  con- 
tact between  the  liquid  and  the  tube.  If  the  tube  is  of 
small  bore,  the  liquid  will  rise  inside  the  tube,  equilibrium 
being  established  when  the  weight  of  the  column  of  liquid 
within  the  tube  (provided  the  bore  is  uniform)  equals  the 
vertical  component  of  the  force  due  to  surface  tension. 
If  d  is  the  density  of  the  liquid  in  gm.  per  cc.,  h  the  height 
in  cm.  of  the  liquid  column  above  the  general  surface  of 
the  liquid  outside  the  tube,  and  g  is  the  acceleration  due 
to  gravity  (approximately  980  cm.  per  sec2.),  it  follows  that 

TT  r2  h  d  g  =  2  TT  r  T  cos  a. 

From  this  equation  the  value  of  T,  the  surface  tension 
in  dynes  per  cm.,  can  be  found. 

(a)  Capillary  tubes  of  different  sizes  are  provided. 
These  may  be  thermometer-tubes  or  larger  glass  tubing 
drawn  out  to  a  fine  bore.  In  either  case  every  precaution 
must  be  taken  to  have  the  tubes  perfectly  clean  and  free 
from  all  traces  of  grease.  They  should  be  cleaned  with 
caustic  potash  solution,  rinsed  with  tap  water  and  then 


26 


CAPILLARITY 


[6 


B 


Fig.  3. — Showing  the  capillary  action  between  water  and  glass  in 
tubes  of  various  shapes  and  sizes. 


with  the  liquid  to  be  experimented  with  (in  this  case 
water).  With  a  rubber  band  fasten  the  tubes  side  by 
side  to  a  glass  scale,  and  place  the  scale  and  tubes  verti- 
cally in  a  small  dish  of  distilled  water.  Lower  the  tubes 
first  to  the  bottom  of  the  dish  so  as  to  wet  the  inside  for 
some  distance  above  the  point  to  which  the  water  will 
rise.  Then  clamp  them  with  the  ends  below  the  surface, 
and  note  on  the  scale  the  point  to  which  the  water  rises 
in  each  tube.  To  obtain  the  reading  for  the  water-sur- 
face in  the  dish,  a  wire  hook  is  provided  which  should  be 
brought  up  so  that  the  point  is  just  even  with  the  surface. 
Then  read  the  height  of  this  point  on  the  glass  scale. 

(b)  Measure  the  inside  diameter  of  the  tube  (if  its 
bore  is  uniform)  with  a  micrometer  microscope  or  by 
means  of  a  thread  of  mercury  drawn  into  the  tube.  In 
case  the  latter  method  is  used,  the  length  and  mass  of 


6-7]  CAPILLARITY  27 

the  thread  and  the  density  of  mercury  are  all  the  data 
needed  for  calculating  the  diameter.  If  the  tube  is  not 
uniform  in  bore,  it  will  be  necessary  to  find  the  diam- 
eter at  the  point  to  which  the  water  rises;  this  can  be 
done  by  breaking  the  tube  by  scratching  it  with  a  file  at  the 
desired  point. 

(c)  In  the  same  way  find  the  surface  tension  of  alcohol. 
For  alcohol  and  glass  the  angle  of  contact  is  practically 
zero. 


(d)  Calculate  the  surface  tension  of  water  from  the 
data  for  each  tube,  and  take  the  average.  For  pure  water 
and  clean  glass  the  angle  of  contact  is  practically  zero. 
Would  the  water  or  alcohol  rise  as  high  in  the  tubes  if 
the  experiment  were  performed  in  a  vacuum?  Explain. 

The  height  at  which  a  liquid  will  be  held  by  capillarity 
in  a  tube  depends  upon  the  diameter  of  the  tube  only  at 
the  upper  end  of  the  liquid  column  where  the  surface 
film  is  attached  to  the  tube,  being  independent  of  the 
shape  and  size  of  the  tube  everywhere  else.  Explain. 


7.     VISCOSITY.     FLOW  OF  LIQUIDS  IN  TUBES. 

To  determine  the  coefficient  of  -viscosity  of  a  liquid. 

When  a  liquid  flows  through  a  tube,  its  motion  is  uni- 
form under  the  action  of  two  opposing  forces,  that  due 
to  the  head,  or  pressure  difference  between  the  two  ends, 
urging  it  forward,  and  that  of  viscosity,  opposing  the 
motion.  The  latter  force  is  proportional  to  the  rate  at 
which  the  liquid  is  forced  through,  to  the  length  of  the 


28  VISCOSITY  [7 

tube,  and  inversely  proportional  to  the  square  of  the 
radius,  (if  the  tube  has  a  circular  cross-section). 

V         I 

F  =  —  x   —   X  constant, 
t          r* 

If  the  tube  is  of  uniform  cross-section,  the  force  urging 
the  liquid  forward  equals  the  pressure  difference  between 
the  two  ends  multiplied  by  this  cross-section.  We  there- 
fore have 

F'  =  p  TT  r2. 

If  the  liquid  moves  with  unaccelerated  motion,  F  =  Ff, 
whence 

~  p  TT  r*  t 

Constant    =  — 

V  I 

In  this  expression,  p  is  the  pressure  difference  between  the 
ends  of  the  tube,  V/t  is  the  volume  flowing  through  per 
second,  /  is  the  length  of  the  tube,  r  its  radius.  If  all 
these  quantities  are  measured  in  C.  G.  S.  units  (pressure 
in  dynes  per  cm.2),  the  constant,  when  divided  by  8,  is 
called  the  coefficient  of  viscosity  of  the  liquid  in  C.  G.  S. 
measure.  The  factor  8  arises  in  the  fact  that  the  coefficient 
of  viscosity  was  not  originally  defined  in  this  way. 

If  the  coefficient  of  viscosity  is  measured  by  determin- 
ing experimentally  all  the  other  quantities  occurring  in 
the  equation  and  is  found  to  be  constant  even  though  the 
conditions  of  the  experiment  are  varied,  you  will  have 
succeeded  in  verifying  the  statements  made  above.  Cap- 
illary glass  tubes  are  used,  bent  in  the  form  of  a  syphon. 

(a)  Fill  one  of  the  syphons  with  tap  water  and  arrange 
it  so  as  to  transfer  the  liquid  from  one  beaker  to  another 
at  a  lower  level.  The  difference  in  level  of  the  surfaces 


7]  VISCOSITY  29 

of  water  in  the  two  beakers  gives  the  required  pressure- 
difference,  provided  that  the  end  of  the  tube  is  below  the 
surface  in  the  lower  beaker  as  well  as  in  the  upper  one. 
If  there  is  an  appreciable  change  in  level  in  the  beakers 
during  the  experiment,  take  the  mean  value  of  the  dif- 
ference. Use  a  head  of  water  of  not  over  3  cm. 

Allow  the  syphon  to  run  for  an  observed  length  of 
time,  and  determine  the  volume  of  liquid  which  runs 
through  by  weighing  the  beakers  and  contents  at  the 
beginning  and  end  of  the  run.  Note  the  temperature  of 
the  water.  Weigh  both  beakers,  and  take  the  mean  of 
the  two  determinations,  in  case  it  is  desirable  to  increase 
the  accuracy  of  the  determination  of  V.  In  this  way 
make  one  determination  with  each  of  three  tubes  of 
different  length  and  cross-section. 

Note.  The  tube  must  be  very  clean.  Also  the  method 
is  only  accurate  if  the  fluid  is  run  through  very  slowly. 

(b)  I  may  be  measured  with  a  paper  scale.       Secure 
data  for  calculation  of  the  radii  of   the   tubes  as  follows. 
Fill  a  part  of  the  tube  with  clean  mercury.      Measure 
the  length  of  the  thread  in  several  different  positions  in 
the  tube,   distributed  over  its   whole  length.      Run  the 
mercury  into  a  small  beaker  or  a  watchglass  and  weigh 
carefully  on  a  sensitive  balance. 

(c)  Make  a  determination  of  the  coefficient  of  viscosity 
of  tap  water  at  about  60°  C. 

The  effect  of  evaporation  may  be  eliminated  by  aver- 
aging loss  in  one  beaker  with  gain  in  the  other,  provided 
the  water  is  at  the  same  temperature  in  both  beakers. 


(d)     Calculate  the  radii  of  the  tubes  from  your  data. 
The  density  of  mercury  at  18°  C.  is  13.55.     Calculate 


30  VISCOSITY  [7-8 

the  coefficient  of  viscosity  for  the  various  cases  under 
test,  expressing  your  results  in  C.  G.  S.  Units  and  present- 
ing them  in  tabular  form. 

(e)  Calculate  the  probable  error  of  your  mean  value 
for  the  coefficient  of  viscosity  for  water  at  room  tempera- 
ture. 

(/)  State  clearly  the  way  in  which  the  rate  of  flow  of 
a  liquid  depends  on  the  length  and  diameter  of  the  tube 
and  the  character  of  the  liquid. 

Since  the  water  in  the  upper  beaker  is  at  rest,  accelera- 
tion must  be  imparted  to  it  in  order  to  cause  it  to  flow 
through  the  siphon.  Our  equation  is  therefore  only 
approximately  true.  Will  the  observed  value  of  the 
constant  be  larger  or  smaller  than  the  true  value  on 
this  account? 

8.     YOUNG'S  MODULUS  BY  STRETCHING. 

References.—  Duff,  pp.  119,  121;  Kimball,  159. 

Hooke's  Law  states  that  in  elastic  bodies,  within  their 
elastic  limits,  the  strain  or  deformation  produced  is  pro- 
portional to  the  change  in  the  stress  or  distorting  force. 
In  particular  it  states  that  if  different  forces  be  applied  to  a 
wire,  e.  g.,  by  suspending  it  and  hanging  masses  from  it, 
the  amount  of  stretching  will  be  (within  certain  limits) 
proportional  to  the  applied  force.  For  a  wire  of  any  given 
material  the  ratio  of  the  change  in  stress  per  unit  area  of 
cross-section  to  the  increase  in  length  per  unit  length  is 
known  as  Young's  modulus.  If  f*  is  the  additional  force 
applied  to  a  wire  of  length  L  and  cross-section  a,  and  /  is 
the  elongation  produced,  the  value  of  the  ratio  is 


/     /         P 

1/1      -        —  /.  —  or  — 
r  I      ~        a        L        al 


8]  YOUNG'S  MODULUS  BY  STRETCHING  31 

It  is  approximately  a  constant  for  any  given  material, 
thus  verifying  Hooke's  law.  The  constant  has  widely 
varying  values,  however,  for  different  materials. 

In  determining  the  stretch  /  of  the  wire  a  spirit  level,  C , 
carried  in  a  hinged  frame  is  used.  The  frame  is  sup- 
ported by  two  wiresA^If  one  of  these  wires  is  stretched 
the  micrometer  sere Wp.  supporting  one  end  of  the  level 
must  be  turned  up  before  the  level  will  read  as  before. 
The  amount  of  stretch  can  thus  be  directly  measured. 

•  * «. 

(a)  Hang  first  a  large  enough  mass  on  the  wire  to 

insure  that  it  is  straight  at  the  beginning.    Make  a  setting     J 
with  the  screw  and  read  it  to  0.01  mm.  or  less.     Then    / 
increase  the  load  by  adding  500  gm.  at  a  time  (each  time   L 
reading  the  screw),  until  the  wire  carries  a  load  of  about 
3000  gm.,  or  until  the  elastic  limit  is  approached. 

Take  the  masses  off,  500  gm.  at  a  time,  reading  the  screw 
each  time. 

(b)  Repeat  (a)   at  least  once.     Record  the  observa- 
tions in  tabular  form. 

(c)  Measure  the  length  of  the  wire  between  the  clamps, 
and  also  the  diameter  of  the  wire.    In  measuring  the  latter, 
apply  the  caliper  to  four  or  five  different  points  along  the 
wire. 


(d)     From  the  data  in  (a)  and  (6)  calculate  separately 
for  each  series  the  mean  elongation  produced  by  a  stretch- 
ing  force  of  500  grams- weight,  as  follows: 

Take  the  difference  between  each  reading  and  every    / 
other  reading  of  the  series,   making  as  many  combina-  ' 
tions  as  possible.     Divide  the  sum  of  these  differences 
by  the  sum  of  the  number  of  500  gm.   weights  which 
produced  them.     For  example,   with   an  initial  reading 


32  YOUNG'S  MODULUS  BY  STRETCHING  [8 

(called  0)  and  four  readings  (called  1,  2,  3,  4)  taken  after 
adding  one,  two,  three,  and  four  weights  respectively 
•  we  have, 

Reading  Difference  No.  of  Difference  No.  of 

Weights  Weights 

0  3.62  0  -  1  =  0.45  1  1  -  3  =  0.90  2 

1  4.07  0  -  2  =  0.92  2  1  -  4  =  1.35  3 

2  4.54  0-3  =.1.35  3  2-3  =  0.43  1 

3  4.97  0  -  4  =  1.80  4  2  -  4  =  0.88  2 

4  5.42  1  -  2  =  0.47  1  3  -  4  =  0.48  1 

Sum   of  differences    =    9.03.      Sum  of   weights    =    20. 

Average  difference  per  500  gm.  weight  =  0.452  mm. 

,  ^ 

Expressing  the  force  in  dynes  and  the  lengths  in  cm.,  | 
calculate  Young's  modulus.  (  ; 


(e)  Should  the  result  be  independent  of  the  diameter 
of  the  wire? 

What  evidence  does  the  experiment  give  for  the  veri- 
fication of  Hooke's  law?  If  there  is  any  variation  from 
the  law,  assign  a  reason  if  you  can. 

The  Optical  Lever 

In  another  form  of  apparatus  the  elongation  /  is  deter- 
mined using  an  optical  lever  and  a  telescope  and  scale. 
The  optical  lever  stands  upon  three  pegs  with  spherical 
ends,  ABC,  two  of  these,  A,  B,  resting  in  a  hole  and 
groove  respectively,  form  the  fulcrum  of  the  lever,  while 
the  third,  C,  rests  on  the  flat  horizontal  surface  of  a  chuck 
H  attached  to  the  wire  under  test.  The  lever  carries  a 
mirror-which  is  adjusted  to  lie  in  a  plane  through  A  B. 

From  the  figure  it  is  seen  that  the  stretch  /  is  given  by 
the  relation  /  =  r  sin  A.  r  may  be  determined  from  the 
distances  between  the  centers  of  the  spheres  A,  B,  C, 


8-9] 


ALCOHOL  AND  WATER  THERMOMETERS 


33 


T-0* 


Fig.  4. 

from  the  relation  r  =  /a2  -c2 .  Where  a  is  the  average  of 
the  distances  AC,  BC  and  c  half  the  distance  AB.  The 
double  angle  2a  is  determined  from  the  radius\R  (distance 
from  the  mirror  to  the  curved  scale)  and  the  length  of  arc 
(the  difference  between  successive  readings^  in  the  tele- 
scope) by  means  of  the  relation  Angle  (in  radians)  x  radius 
=  length  of  arc. 

If  you  used  the  optical  lever,  prove  from  the  law  of 
reflection  that  a  beam  of  light  reflected  by  a  mirror 
turns  through  twice  the  angle  through  which  the  mirror 
turns. 


9.     COMPARISON    OF    ALCOHOL    AND    WATER 
THERMOMETERS. 

In  this  experiment  the  relative  expansions  of  water  and 
alcohol  are  to  be  studied,  and  the  behavior  of  these  liquids 
when  used  in  thermometers  is  to  be  observed. 

(a)  Two  thermometer-bulbs  are  to  be  filled,  one  with 
water  and  the  other  with  ethyl  alcohol,  by  the  aid  of  the 
reservoir-tube.  The  reservoir  is  fitted  on  the  end  of  the 


34  ALCOHOL  AND  WATER  THERMOMETERS  [9 

thermometer-stem,  filled  with  the  liquid  and  warmed. 
(In  the  case  of  the  water-thermometer,  the  water  should 
first  be  boiled  to  drive  out  the  oxygen  held  in  solution, 
before  filling  the  reservoir  with  it.)  The  liquid  is  then 
introduced  into  the  thermometer-bulb  by  alternately 
heating  the  bulb  to  drive  out  the  air  and  allowing  it  to 
cool  to  admit  the  liquid.  When  all  but  a  tiny  bubble  of 
air  has  been  removed,  place  the  bulb  in  ice-water  and 
force  the  liquid  to  dissolve  the  air.  If  this  does  not  suc- 
ceed, ask  for  assistance.  Take  care  not  to  ignite  the 
alcohol.  The  liquid  in  each  thermometer  should  stand 
1  or  2  cm.  above  the  lower  end  of  the  stem  when  the  bulb 
is  in  melting  ice;  for  this  purpose  it  will  be  found  neces- 
sary to  remove  liquid  from  the  stem  by  means  of  a  fine 
wire. 

(6)  Glue  or  otherwise  fasten  a  strip  of  stiff  paper  along 
the  back  of  each  stem,  and  use  it  as  a  scale.  Then  place 
the  thermometers  in  clamps  with  their  bulbs  in  shaved 
ice  or  in  a  mixture  of  water  and  ice.  When^the  readings 
become  steady,  indicate  the  position  of  the  meniscus  of 
each  by  a  sharp  line  on  the  card.  Mark  the  line  zero. 
This  is  the  first  fixed  point  of  the  thermometer. 

(c)  To  determine  a  second  fixed  point,  place  the  bulbs 
in  a  beaker  of  methyl  alcohol  which  is  itself  placed  on  a 
support  in  a  bath  of  water.     Heat  the  water-bath  slowly 
until  the  methyl  alcohol  begins  to  boil.     Be  very  careful 
not  to  bring  the  alcohol  itself  to  the  flame,  and  avoid 
inhaling  the  fumes.     When  the  readings  become  steady, 
again  indicate  the  position  of  the  meniscus  on  each  stem 
by  a  sharp  line.    Mark  this  point  66,  which  is  the  boiling 
point  of  methyl  alcohol  on  the  centigrade  scale. 

(d)  Lay  the  stems  of  the  thermometers  on  a  flat  sur- 
face, measure  the  distance  on  each  between  the  two  fixed 


9]  ALCOHOL  AND  WATER  THERMOMETERS  35 

points,  then  by  means  of  the  customary  geometrical 
construction  divide  this  distance  into  66  equal  parts  and 
call  each  part  a  degree.  Put  the  marks  for  each  degree 
on  the  scale  and  number  every  tenth  one. 

(e)  Place  the  two  arbitrarily  calibrated  thermometers 
in  a  water-bath  at  0°,  as  recorded  by  each  thermometer. 
Gradually  raise  the  temperature  of  the  water-hath  and 
note  the  readings  of  the  two  thermometers,  at  first  at 
short  intervals,  then  at  longer  intervals,  until  the  upper 
fixed  point  is  reached.  Each  time  that  the  temperature  is 
raised,  it  should  be  kept  at  its  new  value  for  three  or  four 
minutes,  so  as  to  give  the  bulbs  time  to  assume  the  tem- 
perature of  the  bath. 


(/)  Plot  the  series  of  points  on  co-ordinate  paper,  hav- 
ing as  abscissae  the  temperatures  by  the  alcohol-ther- 
mometer, and  as  ordinates  the  corresponding  temperatures 
by  the  water-thermometer.  Draw  a  smooth  curve  through 
the  points.  This  curve  gives  the  relation  between  the 
temperatures  as  recorded  by  the  two  thermometers. 
For  the  general  method  of  plotting  data,  see  the  direc- 
tions, page  41. 

What  inferences  can  you  draw  from  the  curve?  If 
alcohol  be  taken  as  the  standard  substance,  what  can 
you  say  of  the  uniformity  of  the  expansion  of  the  water? 
If  water  be  assumed  as  the  standard,  what  of  the  expan- 
sion of  the  alcohol?  Which  would  be  the  better  sub- 
stance to  use  in  a  practical  thermometer,  and  why? 


36    EXPANSION  OF  LIQUID  -REGNAULT'S  METHOD    [10 

10.    COEFFICIENT  OF  EXPANSION  OF  A  LIQUID 
BY  REGNAULT'S  ABSOLUTE  METHOD. 

To  determine  the  coefficient  of  expansion  of  a  sample  of  oil. 

If  the  expansion  coefficient  of  a  liquid  is  determined  by 
measuring  its  volume  at  different  temperatures,  correc- 
tion must  be  made  for  the  expansion  of  the  vessel  used 
in  the  volume  measurements.  The  present  method  was 
devised  by  Dulong  and  Petit  for  the  purpose  of  eliminating 
that  correction.  It  was  later  improved  and  made  practical 
by  Regnault. 

The  apparatus  consists  essentially  of  a  metal  U-tube. 
One  arm  is  surrounded  by  a  steam-jacket,  the  other  by 
an  ice-pack.  Each  arm  terminates  in  a  glass  tube,  so 
that  the  heights  of  the  liquid  surfaces  may  be  read  off. 

The  oil  in  the  hot  side  is  less  dense  than  that  on  the 
cold  side,  and  hence  stands  higher.  Since  the  pressure  at 
the  bottom  of  the  two  sides  is  the  same,  we  have 

h0  d0  g  =  ht  dt  g,  (1) 

where  h0  is  the  total  height  of  the  cold  column,  d0  the 
density  of  the  cold  oil,  ht  and  dt,  the  values  of  the  same 
quantities  on  the  hot  side,  and  g  is  the  number  of  dynes 
in  a  gram.  The  last  factor  is  inserted  in  order  to  give 
the  pressures  in  C.  G.  S.  units. 

Now  if  V0  is  the  volume  of  a  given  mass  of  oil  at  0°  C., 
and  Vt  its  volume  at  t°  C.,  we  have 

V,  =  V.  (1  4-  j8<),  (2) 

j8  being  the  cubical  expansion  coefficient  of  the  oil.  Since 
the  densities  are  inversely  as  the  volumes,  this  gives 

d0  I  dt  =  1  +  #  (3) 


10]    EXPANSION  OF  LIQUID— REGNAULTS  METHOD    37 

Combining  this  with  the  first  equation  above  gives 

(A,  -  *.)  /  *  fc.  -  ft  (4) 

From  this,  /3  may  readily  be  determined  by  observation 
of  ht,  h0,  and  t. 

(a)  With  the  two  arms  -of  the  manometer  at  the  same 
temperature,  read  the  level  of  the  oil  in  the  glass  tubes. 
If  the  two  levels  are  not  the  same,  it  may  be  due  either 
to  the  instrument  not  being  level,  or  to  the  presence  of 
foreign  material  in  the  oil  (air,  water).  Adjust  until 
the  two  levels  show  the  same  reading  before  proceeding. 

(6)  The  manometer  should  contain  oil  enough  so  that 
when  the  temperature  difference  is  established,  the  level 
on  the  cold  side  can  be  conveniently  observed.  The 
column  projecting  above  the  ice-pack  on  one  side  and 
the  steam  jacket  on  the  other  should  be  as  short  as 
possible,  as  the  formula  is  based  on  the  assumption  that 
the  whole  of  each  arm  is  at  one  single  temperature. 

Pack  the  ice  jacket  full  of  shaved  ice  and  keep  it  full. 
Let  the  water  run  off  below  and  keep  the  ice  well  packed 
down.  There  is  a  tendency  for  the  lower  part  of  the 
cold  side  to  warm  up,  by  conduction  from  the  hot  side. 
This  is  partly  prevented  by  the  insertion  of  a  non-con- 
ducting segment  in  the  cross-tube.  But  if  the  ice  is  not 
packed  down,  it  will  melt  out  below  and  introduce  error. 

Pass  steam  through  the  steam  jacket,  from  above 
downward.  The  steam  should  escape  freely  at  the  lower 
opening,  and  not  all  pass  out  at  the  small  opening  around 
the  glass  tube.  While  heating,  slide  the  cap  up  so  as  to 
cover  the  oil  column  completely. 

(c)  In  reading,  slide  the  cap  on  the  steam  jacket 
down  below  the  meniscus  far  enough  to  read  conveniently. 


38         EXPANSION  OF  A  LIQUID.     PYCNOMETER     [10-11 

Your  first  reading  may  be  in  error  if  time  has  not 
been  allowed  for  temperatures  to  become  steady.  A 
point  will  be  reached,  however,  beyond  which  further 
variations  are  irregular,  no  uniform  increase  being 
observed.  Now  let  out  a  little  oil  by  means  of  the  stop 
cock  at  the  bottom  of  the  U-tube  and  read  both  columns 
again.  Repeat  this  process  at  least  twice. 

(d)     Calculate  the  value  of  ft. 


11.    COEFFICIENT  OF  EXPANSION  OF  A  LIQUID 
BY  PYCNOMETER  METHOD. 

To  determine  the  coefficient  of  cubical  expansion  of  alcohol. 

The  method  consists  in    finding    the    mass  of  alcohol 
filling  the  pycnometer  at  each  of  several  temperatures. 

Let   Mo  represent   the   mass   of   alcohol   which   would 
fill  the  pycnometer  at  0°  C., 

V0,  the  volume  of  M0,  and  hence  of  the  pycnometer 
at  0°  C., 

m,  the  difference  between  the  masses  contained  in  the 
pycnometer  at  0°  C.  and  at  t°  C., 

ft,  y,  the  coefficients  of  cubical  expansion  respectively 
of  alcohol  and  glass. 

Now 

Fo   (1  +  fti)   =  the  volume  of  the  mass  M0  at  t°  C,  and 
V0  (1  +  yt)  =  the  volume  of  the  pycnometer  at  t°  C,  and 


11]          EXPANSION  OF  A  LIQUID.     PYCNOMETER          39 

V0  (1  +  fit)  -  V0  (1  +  7  i)  =  V0  (/3  -  7)  t  =  the  volume 
of  the  mass  m  at  t°.  If  d0  and  dt  represent  the  densities 
of  liquid  at  0°  and  t°,  respectively,  then 

(1)  d0  =  Mo/Fo, 

(2)  dt  =  m/V0  (0  -  y)t, 

(3)  d0  =  dt(l  +  ftt)       (See  Exp.  10). 

By  eliminating  d0  and  dt  from  equations  (1),  (2),  and  (3), 
we  get 

(4)  Mo  (0-7)*  =  w(l  +  0*). 

Besides  containing  the  known  masses,  MQ  and  m,  this 
equation  contains  the  three  quantities,  0,  7,  and  t.  Any 
two  of  these  three  quantities  being  known,  the  third  will 
be  given  by  the  equation. 

Solving  this  equation  for  0  we  have 

0  =  m  I  (M0  -  m)  t  +  Mo  7  /  (M0  -  m). 

(a)  Fill  the  pycnometer  with  alcohol  and  set  it  on  a 
platform  in  a  kettle  of  water,  so  that  the  water  comes  well 
up  to  the  neck  of  the  pycnometer.    Hang  a  50°  thermome- 
ter in  the  bath  alongside  the  pycnometer,  and  keep  the 
bath  well  stirred  for  about  five  minutes.     The  tempera- 
ture of  the  bath,  which  must  be  a  little  above  that  of  the 
room,   should   remain   constant   within   0.1°    during   this 
time,  and  at  the  end  of  it  the  alcohol  will  have  the  same 
temperature  within  0.1°.     Take  the  pycnometer  out  of 
the  bath,  wipe  the  outside  dry,  and  weigh   to    an  ac- 
curacy of  1  mg. 

(b)  In  this  way  take  at  least  four  readings,  at  tem- 
peratures ranging  up  to  50°  C.     Keep  the    temperature 
steady  in  each  case  for  five  or  ten  minutes  by  holding  the 
flame  under  the  kettle  for  a  few  seconds  occasionally. 


40          EXPANSION  OF  A  LIQUID.     PYCNOMETER         [11 

Before  reheating  after  each  weighing,  refill  the  pycnometer. 
This  will  prevent  the  formation  of  a  bubble  under  the 
stopper. 

(c)     Empty  the  alcohol  into  the  bottle  from  which  it 
was  taken,  dry  the  pycnometer,  and  weigh. 


(d)  Determine   the    mass    of   the    alcohol   filling   the 
pycnometer  at  each  temperature.     Plot  the  results,  with 
temperatures,  starting  from  0°,  as  abscissae,  and  masses 
as  ordinates.     Assuming  that  the  expansion  is  uniform, 
draw  the  straight  line  which  best  represents  the  plotted 
points,  and  by  extending  it  find  the  mass  filling  the  pyc- 
nometer  at  0°C,   and   at   50°C,   and   then   calculate   the 
coefficient  of  expansion  of  alcohol. 

(e)  Error   in   this   experiment   may   be   estimated   by 
carrying  out  the  calculation  of  /3  separately  for  each  pair 
of  values  of  m  and  t,  and  comparing  the  results. 


PLOTTING  OF  CURVES  41 

PLOTTING  OF  CURVES. 

It  is  very  frequently  advantageous  to  represent  the 
way  in  which  one  quantity  depends  on  another  by  means 
of  a  plot,  or  graph.  The  values  of  the  quantity  regarded 
as  independently  variable  are  laid  off  on  the  horizontal 
axis,  and  are  called  abscissa.  The  corresponding  values 
of  the  dependent  variable  are  called  ordinates,  and  are 
laid  off  on  the  vertical  axis. 

Choose  the  scale  of  abscissas  so  that  the  largest  value 
reaches  nearly  across  the  page.  Choose  the  scale  of 
ordinates  so  that  the  largest  value  reaches  nearly  the 
vertical  distance  assigned  to  the  curve.  Always  use 
scales  such  that  each  space  of  the  co-ordinate  paper 
represents  a  convenient  number  of  units,  or  a  convenient 
fractional  part  of  a  unit.  The  two  scales  need  not  be  alike. 

As  a  rule,  take  for  origin  of  co-ordinates  the  point  in 
the  lower  left-hand  corner  of  the  sheet  of  plotting  paper, 
and  for  co-ordinate  axes  the  two  perpendicular  lines 
which  intersect  at  this  point.  Along  these  axes  indicate 
the  scales  used. 

Each  pair  of  values  plotted  should  be  plainly  indicated 
by  a  point  at  the  center  of  a  small  circle,  or  by  a  cross 
intersecting  at  the  point  plotted. 

After  the  values  are  plotted,  draw  the  smooth  curve 
that  best  fits  all  the  points  plotted.  This  curve  will  not, 
in  general,  pass  through  all  the  points.  Deviation  of 
points  from  this  curve  usually  indicates  errors  of  obser- 
vation. 

Give  each  curve  a  title. 

All  curves  should  be  drawn  on  the  best  18  x  25  mm. 
co-ordinate  paper.  Both  curves  and  lettering  should  be 
done  in  ink. 


42     EXPANSION  OF  GLASS.  WEIGHT  THERMOMETER    [12 

12.      COEFFICIENT   OF   EXPANSION    OF    GLASS 
BY  WEIGHT-THERMOMETER  METHOD. 

To  determine  the  coefficient  of  cubical  expansion  of  glass 
by  means  of  the  weight-thermometer. 

The  weight-thermometer  consists  of  a  glass  tube  closed 
at  one  end  and  ending  in  a  curved  capillary  at  the  other 
end.  It  is  filled  with  mercury  at  0°C.,  and  the  mass  of 
the  mercury  measured.  When  later  placed  in  a  bath  of 
higher  temperature,  some  mercury  overflows,  since  mer- 
cury expands  more  rapidly  when  heated  than  does  glass. 
The  mass  of  this  overflow  is  measured. 

Equation  4,  as  developed  in  the  previous  experiment, 
applies  here  also,  but  M0  and  m  are  measured  directly. 
13  and  t  are  to  be  observed  and  y  calculated. 

(a)  Weigh  the  empty  weight-thermometer  to  an 
accuracy  of  10  mg.  Then  fill  it  with  mercury.  In  doing 
so  it  should  be  held  by  a  clamp,  or  suspended  in  a  gauze 
jacket,  and  heated  by  a  flame  held  in  the  hand,  care  being 
taken  to  keep  from  heating  too  rapidly  and  from  apply- 
ing the  flame  too  long  at  any  point  of  the  empty  bulb. 

The  end  of  the  capillary  dips  under  the  surface  of  mer- 
cury in  a  porcelain  dish.  The  mercury  in  this  dish  should 
first  be  heated,  and  then  the  weight-thermometer  gently 
heated  until  the  air  bubbles  out  through  the  mercury. 
On  allowing  the  bulb  to  cool,  some  mercury  will  run  into 
it.  The  process  is  then  repeated.  When  considerable 
mercury  is  in  the  bulb,  heat  it  until  it  boils  vigorously, 
but  be  careful  not  to  heat  too  hot  that  portion  of  the  glass 
where  there  is  no  mercury.  Keep  the  mercury  hot  in  the 
porcelain  dish,  otherwise  the  glass  is  apt  to  crack  when  the 
cooler  mercury  rushes  in.  The  tube  must  be  completely 
filled  with  mercury  to  the  end  of  the  capillary,  the  last 


12]  EXPANSION  OF  GLASS.    WEIGHT  THERMOMETER  43 

bubble  of  air  being  expelled.  To  accomplish  this  it  will 
be  found  helpful  to  turn  the  weight-thermometer  so  as  to 
give  the  capillary  above  the  air-bubble  an  upward  slant. 
A  gentle  tapping  with  a  light  splinter  or  pencil  will  then 
probably  cause  the  bubble  to  work  its  way  along  the  tube 
far  enough  to  be  easily  expelled  by  further  heating. 

(b)  Keeping  the  end  of  the  capillary  in  the  dish  of 
mercury,   allow   the  weight-thermometer  to   cool  in  the 
air  sufficiently  so  that  you  can  bear  your  hand  on  it,  and 
then  surround  it  with  shaved  ice  and  leave  it  long  enough 
to  contract  as  much  as  it  will.    Assume  that  its  tempera- 
ture is  now  0°C.     Carefully  remove  the  dish  and  brush 
the  mercury  off  the  end  of  the  capillary.     Place  a  watch- 
glass  under  the  end  to  catch  the  mercury  as  it  begins  to 
expand  and  flow    out.      Now  remove  the  ice-bath  and 
warm  the  bulb  with  the  hand  until  its  temperature  is 
raised  to  the  temperature  of  the  room. 

(c)  Place  the  weight-thermometer  in  the  boiler  pro- 
vided.    Heat  it  to  the  boiling  point  of  water  by  passing 
steam  over  it  until  no  more  mercury  comes  out.     Read 
the    barometer    and    calculate    the    temperature    of    the 
steam.     Very  carefully  weigh  the  mercury  in  the  watch- 
glass  to  an  accuracy  of  1   mg.     Weigh  the  weight-ther- 
mometer and  contained  mercury  to  an  accuracy  of  10  mg. 


(d)  From  the  data  obtained  in  (a)  and  (c)  determine 
M0  and  m  Using  your  values  of  M0  and  m,  and  taking 
the  coefficient  of  expansion  of  mercury  as  found  in  the 
Tables,  calculate  the  coefficient  of  cubical  expansion  of 
glass. 

What  additional  measurements  would  you  need  to 
have  made  in  order  to  measure  the  room  temperature 
with  your  weight- thermometer? 


44  CALORIMETRY  [12 

GALORIMETRY. 

Calorimetry  is  that  branch  of  physical  measurements 
which  has  to  do  with  quantities  of  heat.  In  such  changes 
as  temperature-rise,  fusion,  vaporization,  combustion, 
and  solution,  quantities  of  heat  are  either  absorbed,  or 
evolved,  or  transferred  from  one  set  of  substances  to 
another.  The  unit  employed  in  their  measurement  is 
the  calorie. 

Measure  of  heat.  In  the  case  of  the  rise  in  tem- 
perature of  a  body,  the  quantity  of  heat  which  passes 
into  the  body  is  measured  by  the  product  ms(t2—ti), 
where  m  is  the  mass,  5  the  specific  heat,  and  (t*—  t^  the 
rise  in  temperature.  This  is  not  true  if  the  body  changes 
state  during  the  heating.  In  such  a  case  the  quantity  of 
heat  absorbed  during  the  change  of  state  alone  is  meas- 
ured by  the  product  m  L,  where  m  is  the  mass  and  L  the 
heat  absorbed  per  gram.  L  is  called  the  heat  of  the  change 
of  state  (e.  g.,  fusion,  vaporization). 

In  every  experiment  performed  with  a  calorimeter 
the  quantity  desired  is  obtained  by  solving  an  equation 
which  expresses  the  fact  that  the  total  amount  of  heat 
lost  by  those  bodies  which  lose  heat  equals  the  total 
amount  of  heat  received  by  those  bodies  which  receive 
heat. 

The  Calorimeter.  The  vessel  in  which  the  transfer 
of  heat  takes  place,  in  experiments  involving  the  meas- 
urement of  heat,  is  called  a  calorimeter.  It  consists  of 
an  inner  chamber,  insulated  as  completely  as  possible, 
so  as  to  prevent  the  transfer  of  heat  either  to  or  from  the 
material  in  the  chamber.  Transfers  of  heat  by  conduction 
are  eliminated  by  dead  air  or  vacuum  jackets.  Radiation 
is  reduced  by  polished  surfaces.  In  the  Dewar  flask, 
and  the  thermos  bottle,  convection  is  also  eliminated. 


12]  CALORIMETRY  45 

In  precision  work  with  a  calorimeter,  allowance  must 
be  made  for  the  heat  absorbed  by  the  calorimeter  itself, 
with  its  accessories,  thermometer,  stirrer,  etc.  It  is 
convenient  to  calculate  this  correction  in  terms  of  water- 
equivalent.  A  body's  water-equivalent  is  the  mass  of 
water  whose  heat  capacity  is  equal  to  that  of  the  body. 
It  may  be  found  either  by  multiplying  the  body's  mass 
by  its  specific  heat,  or  by  experiment. 

The  water  equivalent  of  a  thermos  bottle  or  Dewar 
flask  may  be  determined  as  follows:  Weigh  the  thermos 
bottle  when  empty,  add  about  25  grams  of  hot  water; 
determining  the  exact  amount  by  re-weighing  after  the 
water  has  been  put  into  the  bottle.  Insert  the  stopper 
with  the  thermometer  and  move  the  bottle  about  so 
that  the  water  will  come  into  contact  with  all  parts  of 
the  inside.  In  particular  be  sure  to  invert  the  bottle 
and  move  it  about  for  some  time  in  this  position  so  that 
all  parts  will  come  to  a  uniform  temperature.  Do  not 
shake  the  bottle. 

Next  pour  in  from  40  to  50  grams  of  water  at  a  known 
temperature  near  that  of  the  room.  Determining  the 
exact  amount  by  weighing  after  pouring  it  in.  Move  the 
flask  about  as  before  and  note  the  constant  temperature 
finally  reached.  Equating  the  heat  lost  by  the  bottle, 
hot  water,  and  thermometer  to  that  gained  by  the  cold 
water  will  give  the  water  equivalent  of  the  bottle  and 
thermometer. 

If  a  heating  coil  is  to  be  used  with  the  thermos  bottle 
its  water  equivalent  may  be  included  in  the  determination 
by  keeping  it  in  the  bottle  throughout  the  preceedure 
just  described. 

In  the  case  of  a  thermometer,  which  is  part  glass  and 
part  mercury,  the  water-equivalent  may  be  determined 


46          SPECIFIC  HEAT.     METHOD  OF  HEATING        [12-13 

by  finding  the  volume  of  the  immersed  part  of  the  ther- 
mometer and  then  calculating  the  water-equivalent  of 
this  volume  of  mercury.  This  is  possible  since  equal 
volumes  of  glass  and  mercury  have  practically  the  same 
heat  capacity,  and  hence  the  thermometer  may  be  treated 
as  though  it  were  made  entirely  of  mercury.  The  student 
will  find  that  0.45  is  approximately  the  factor  by  which 
the  volume  in  cc.  of  the  immersed  part  should  be  multi- 
plied to  give  the  water-equivalent  of  the  thermometer. 

13.     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD 
OF  HEATING. 

To  find  the  specific  heat  of  a  given  liquid. 

In  this  experiment  a  heating  coil,  composed  of  high 
resistance  metal  through  which  an  electric  current  is 
passed,  is  immersed  for  a  given  time,  first  in  water  and 
then  in  the  liquid  whose  specific  heat  it  is  desired  to  find. 
If  -the  same  current  passes  through  the  coil  in  the  two 
cases,  equal  quantities  of  heat  should  be  generated  in 
equal  times.  Noting  in  each  case  the  mass  of  the  liquid 
and  the  rise  in  temperature,  the  two  quantities  of  heat 
may  be  equated  and  the  specific  heat  of  the  liquid  calcu- 
lated, if  that  of  water  is  known. 

(a)  Place  the  bottle,  containing  the  second  liquid,  in 
a  vessel  of  ice-water  to  cool  for  part  (b).  Put  enough 
water  (chilled  to  6°  or  7°  below  room  temperature)  to 
cover  the  heating  coil  to  a  depth  of  about  2  cm.,  deter- 
mining the  exact  amount  by  weighing  the  bottle  both 
before  and  after  the  water  is  poured  in.  Move  the 
bottle  about  so  that  the  water  comes  in  contact  with 
all  parts  of  the  inside  holding  it  inverted  part  of  the 
time.  Note  the  temperature  after  it  becomes  constant. 
Turn  on  the  current  at  a  noted  time  and  let  it  run  for 


13]  SPECIFIC  HEAT.     METHOD  OF  HEATING  47 

exactly  two  minutes.  Equalize  the  temperature  as 
before  and  record  its  value.  Repeat  for  a  series  of  two 
minute  intervals  until  the  temperature  of  the  water  is 
as  far  above  room  temperature  as  it  started  below  it. 

(b)  Repeat   (a),   using  the  second  liquid,   instead   of 
water,  in  the  calorimeter  cup.     It  should  be  chilled  to  a 
lower  temperature  than  the  water,  as  it  will  warm  up 
faster    than    the    water    did.      Heat    for    approximately 
the  same  length  of  time  as  in  (a).     Use  a  considerably 
larger  quantity  of  the  second  liquid  than  you  did  of  the 
water.     It  will  be  necessary  to  find  the  water-equivalent 
of  the  calorimeter  cup,   stirrer,   and  thermometer.     For 
this  purpose  see  the  paragraph  on   Water-Equivalent  on 
page  45. 

(c)  Repeat  (a)  and  (b)  in 'detail. 


(d)  Plot  on  one  sheet  of  co-ordinate  paper  the  results 
of  (a)  and  (b),  using  temperatures  as  ordinates  and  times 
as  abscissae.     Erect  two  perpendiculars  to  the  time  axis 
which  will  include  between  them  as  wide  segments  of  the 
two  curves  as  is  consistent  with  accuracy.     From  these 
intersections   obtain   the   range   of   temperatures   passed 
through  by  the  water  and  the  other  liquid  in  equal  times. 

Write  the  equation  expressing  the  fact  that  the  same 
amount  of  heat  is  absorbed  in  the  two  cases. 

Plot  (c)  also.  The  two  plots  may  be  drawn  on  the 
two  halves  of  the  same  sheet  of  co-ordinate  paper.  From 
the  data  obtained  calculate  the  value  of  the  specific  heat. 

(e)  By  comparing  the  two  determinations,  estimate 
the  probable  error  of  your  result. 

Explain  why  no  correction  is  necessary  for  the  amount 
of  heat  lost  by  radiation. 


48  SPECIFIC  HEAT.     METHOD  OF  COOLING  [14 

14.     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD 
OF  COOLING. 

To  find  the  specific  heat  of  a  given  liquid. 

The  method  consists  in  a  comparison  of  the  quantities 
of  heat  lost  by  two  liquids,  one  of  which  is  water,  when 
equal  volumes  are  allowed  to  cool  under  exactly  the  same 
conditions  through  the  same  range  of  temperature.  The 
conditions  of  radiation  being  the  same  for  both,  if  water 
of  mass  mi  and  specific  heat  Si  cools  through  a  certain 
temperature-range  A£  in  TI  seconds,  and  a  second  liquid  of 
mass  mz  and  specific  heat  s2  requires  T2  seconds  for  the 
same  temperature-change,  then  the  quantities  of  heat  lost 
will  be  proportional  to  the  times,  i.  e.,  Qi/Qz  =  Ti/Tz. 
For  the  similarity  of  conditions  make  the  rate  of  losing 
heat  the  same  in  the  two  cases.  If  w  denote  the  water- 
equivalent  of  the  containing  vessel,  thermometer,  and 
stirrer,  the  above  relation  becomes 


(mi  Si  +  w  Si)  A  J/(m2  s2  +  w  Si)  A  t  = 
from  which  the  unknown  specific  heat  can  be  determined. 

(a)  Set  up  the  calorimeter,  with  the  inner  cup  sup- 
ported from  the  screws  on  the  lower  side  of  the  wooden 
cover.  The  small  nickeled  brass  bottle  is  to  be  supported 
by  the  thermometer,  which  fits  tightly  in  the  cork  of  the 
bottle.  A  second  cork  on  the  thermometer  fits  trie  hole 
in  the  wooden  cover,  and  carries  the  weight  of  the  bottle 
and  contents.  The  bottle  is  thus  suspended  in  air,  and 
cools  primarily  by  radiation.  The  space  outside  the  cup 
should  be  filled  with  water.  /vs#  • 

Fill  the  bottle,  heat  nearly  to  100°C.,  in  a  water-bath, 
handling  the  bottle  by  means  of  the  thermometer.  Trans- 


14] 


SPECIFIC  HEAT.     METHOD  OF  COOLING 


49 


fer  to  the  cooling  chamber,  and  read  the  temperature  at 
one  minute  intervals  until  it  has  fallen  through  a  sufficient 
range  to  give  a  smooth  curve  covering  an  interval  of 
20°  to  30°C.  A  smooth  curve  can  only  >  be "  obtained  if 
the  liquid  is  continuously  and  uniformly  stirred. 

Make  the  necessary  weighings.  The  water-equivalent 
may  be  found  as  explained  under  Calorimetry. 

(b)  Repeat,  using  the  other  liquid.  The  water  in  the 
jacket  should  have  the  same  initial  temperature  as  in 
the  first  case. 


10  20  JO  40          50 

Fig.  5. — Cooling  curves  of  water  and  turpentine. 

(c)  Plot  the  cooling  curves  of  oil  and  water  on  co- 
ordinate paper,  using  temperatures  as  ordinates  and 
corresponding  times  as  abscissae.  (See  Fig.  5.)  Since 
the  temperature  range  must  be  the  same  in  the  two  cases, 
draw  two  lines  parallel  to  the  axis  of  abscissae,  marking 


50  HEAT  OF  FUSION  BY  METHOD  OF  MIXTURES  [14-15 

off  equal  temperature  intervals  on  the  two  curves.  Make 
this  temperature  interval  as  long  as  possible,  consistant 
with  accuracy.  The  intersections  of  these  lines  with  the 
curves  will  give  >the  times  required.  Calculate  the  specific 
heat  of  the  oil. 

If  the  water  had  not  been  changed  between  the  two 
sets  of  observations,  in  what  way  would  the  value  for  the 
specific  heat  of  the  oil  have  been  affected?  What  source 
of  error  still  remains  even  if  the  water  in  the  jacket  is 
changed  before  the  second  measurement?  Suggest  a 
way  to  avoid  this  uncertainty. 


15.     HEAT  OF  FUSION  BY  METHOD  OF 
MIXTURES. 

To  determine  the  heat  of  fusion  of  Wood's  Alloy  by  the 
method  of  mixtures. 

The  melted  alloy,  with  container,  is  plunged  into  water 
in  a  calorimeter  cup.  The  following  changes  occur 
wherein  heat  is  given  out:  (1)  The  alloy  cools  as  a  liquid 
from  the  temperature  of  the  hot  water-bath  down  to  the 
freezing  point  of  the  alloy,  (2)  the  alloy  changes  from  a 
liquid  to  a  solid  without  change  of  temperature,  (3)  the 
alloy  cools  as  a  solid  from  its  freezing  point  down  to  the 
final  temperature  of  the  mixture  in  the  calorimeter,  (4) 
meanwhile  the  nickle  crucible  and  the  copper  cage  cool 
from  the  temperature  of  the  hot  water-bath  to  the  final 
temperature  of  the  mixture.  The  changes  wherein  heat  is 
absorbed  are  those  accompanying  the  rise  in  temperature 
of  the  nickeled  brass  calorimeter  cup  and  contents  form 
the  initial  temperature  of  the  cold  water  up  to  the  final 
temperature  of  the  mixture.  From  these  data,  if  the 


15]  HEAT  OF  FUSION.     WOOD'S  ALLOY  51 

specific  heat  of  the  metal  in  the  solid  and  in  the  liquid 
state  be  known,  the  heat  of  fusion  may  be  found. 

(a)  Make  the   necessary   weighings.      Determine  the 
melting  point  by  melting  the  alloy  and  noting  its  freezing 
point  on  cooling.     The  heating  is  conveniently  done  in 
a  water  bath.     The  thermometer  must  be  placed  in  the 
water,  and  not  in  the  alloy,  as  the  liquid  alloy  wets  the 
glass. 

Set  up  the  calorimeter,  using  the  sensitive  thermometer. 
The  procedure  outlined  under  (d)  gives  the  desired 
result  with  accuracy  only  in  case  the  initial  temperature 
of  the  water  is  the  same  as  that  of  the  room. 

(b)  Bring  the  water  to  the  boiling  point,  taking  care 
that   no   water  gets  inside   the   crucible   containing   the 
alloy.     Remove  the  crucible,  and  quickly  lower  it  into 
the  calorimeter,  right  side  up.     Note  the  time  when  the 
crucible  is  immersed  in  the  cold  water.    Then,  at  the  end 
of  every  5  minutes  for  15  minutes,  read  the  temperature 
of  the  mixture,  keeping  it  well  stirred  meanwhile. 

Do  not  leave  the  gas  burning  near  the  calorimeter. 

(c)  Repeat  (a)  and  (b)  to  check  your  work. 


(d)  Let  M  be  the  mass  of  the  alloy;  m,  the  mass  of 
the  nickle  crucible;  W,  the  mass  of  the  water  in  the 
calorimeter  cup  plus  the  water-equivalent  of  the  cup, 
thermometer,  and  stirrer;  si,  the  specific  heat  of  the 
liquid  alloy;  s2,  that  of  the  solid  alloy;  s3,  that  of  the 
nickel  crucible;  s4,  that  of  the  copper  cage;  T,  the  melt- 
ing point  of  the  alloy;  ti,  the  initial  temperature  of  the 
alloy  and  crucible;  to,  the  initial  temperature  of  the 
calorimeter  and  contents;  /,  the  final  temperature;  and 
L,  the  heat  of  fusion  per  gram  of  the  alloy.  Write  the 


52 


HEAT  OF  FUSION.     WOOD'S  ALLOY 


[15 


proper  equation  representing  the  transfer  of  heat  in  the 
above  process,  using  the  symbols  indicated,  and  solve 
the  equation  for  L.  Take  the  value  of  the  specific  heat 
of  substances  involved  from  the  table  on  page  65. 

From  each  set  of  data  calculate  the  heat  of  fusion 
of  the  alloy,  and  take  the  average.  In  order  to  obtain 
for  each  set  the  value  of  t,  the  temperature  of  the  mixture 
after  the  immersion  of  the  alloy,  proceed  as  follows: 
Plot  the  times  as  abscissae  and  the  temperature  as  or- 
dinates,  and  draw  the  straight  line  which  most  nearly 
represents  the  plotted  points.  This  straight  line  gives  the 
rate  of  cooling  of  the  mixture,  and  will,  when  prolonged 
backward  to  intersection  with  the  axis  of  ordinates,  give 


20* 


i 


0  S  10  IS 

Fig.  6. — Showing  the  method  of  finding  the  radiation  correction. 


15-16]  FUEL  VALUE  OF  ALCOHOL  53 

the  temperature  which  the  mixture  would  have  had  if 
its  temperature  had  been  made  uniform  the  instant  that 
the  alloy  entered  it. 

(e)  Calculate  the  probable  error  of  your  result. 
Point  out  the  principal  sources  of  error  in  the  experiment. 
Examine  the  equation  set  up  in  (d)  and  state  why  only 
a  rough  determination  of  the  melting  point  of  the  alloy 
is  necessary? 


16.     FUEL  VALUE  OF  ALCOHOL. 

To  find  the  amount  of  heat  generated  by  the  combustion 
of  unit  mass  of  the  sample  furnished. 

The  Junker  calorimeter  is  used  (Fig.  7).  This  consists 
of  a  double-walled  copper  vessel  provided  with  an  inner 
cavity  C  for  running  water  and  with  a  number  of  tubes 
or  flues  FF  for  the  heated  gases  from  the  flame.  The 
heat  given  up  by  the  gases  during  their  passage  through 
these  tubes  passes  by  conduction  into  the  stream  of 
water  which  flows  continuously  through  the  calorimeter 
from  the  inlet  /  to  the  outlet  O.  A  thermometer  T2 
placed  at  I  registers  the  temperature  of  the  water  entering 
the  calorimeter  and  a  second  thermometer  T\  at  0  registers 
the  temperature  of  the  water  leaving  it.  A  steady  flow 
of  water  is  maintained  by  means  of  an  overflow  reservoir 
placed  above  the  level  6f  the  calorimeter  and  kept  filled 
with  faucet-water.  The  warm  water,  as  it  issues  from 
the  outlet  O,  is  caught  in  a  beaker  and  weighed. 

The  heat  gained  by  the  water  is  measured  by  the  pro- 
duct m  s  t,  where  ra  is  the  mass  of  water  caught  in  a  given 
time,  s  its  specific  heat,  and  t  the  rise  in  temperature  be- 
tween I  and  O.  This  is  only  an  approximate  measure  of 


54 


FUEL  VALUE  OF  ALCOHOL 


[16 


the  heat  liberated  by  the  combustion  of  the  gas  since 
some  heat  is  lost  by  conduction  and  radiation  while  the 
water  is  flowing  through  the  calorimeter. 

(a)  Arrange  the  calorimeter  as  shown  in  the  figure. 
Place  the  overflow  reservoir  in  position  so  that,  when 
connected  with  the  orifice  I,  a  steady  flow  of  water  through 
the  calorimeter  will  be  established.  For  this  purpose  it 
will  be  found  necessary  to  keep  the  level  of  the  water 
constant.  This  can  be  done  by  adjusting  the  flow  of 


O 


Fig.  7. — Junker's  Calorimeter. 

water  from  the  faucet  to  the  reservoir  in  such  a  way  as 
to  keep  the  water-level  constant.  The  rate  of  flow  is 
conveniently  controlled  by  a  pinch-cock. 

Weigh  the  alcohol  lamp,  after  filling  it,  light  it  at  a 
noted  time  and  place  it  in  position. 

When  the  conditions  have  become  steady,  as  shown  by 
the    constancy    of   the    temperatures    registered    by    the 


16-17]  PRINCIPLE  OF  MOMENTS  55 

thermometers  Ti  and  T2,  place  a  beaker  in  position  to 
catch  the  water  as  it  flows  out  at  0,  and  allow  the  flow  to 
continue  for  a  measured  length  of  time  until  200  gm.  or 
more  of  water  have  passed.  Record  the  time  and  the 
temperature  at  I  and  0. 

Extinguish  the  lamp  at  a  noted  time  and  weigh. 

(6)  Make  two  more  determinations  with  the  rate  of 
flow  of  water  different  each  time. 

(c)  From  the  data  in  (a)  and  (b)  calculate  the  heat 
liberated  per  gram  of  alcohol,  assuming  that  the  rate  of 
consumption  of  alcohol  is  uniform. 

(d)  Calculate  the  probable  error  of  your  result.     To 
what  sources  of  error  is  the  method  of  this  experiment 
subject? 

To  what  sources  of  error  is  the  method  of  this  experi- 
ment subject? 

The  calorimeter  may  also  be  used  to  determine  the 
fuel  value  of  illuminating  gas.  In  this  case,  in  place  of 
the  weight  of  alcohol  consumed,  the  quantities  to  be 
measured  are  volume  of  gas,  corresponding  time,  baro- 
metric pressure  and  gas  pressure. 


17.     THE  PRINCIPLE  OF  MOMENTS:  STATICS. 

To  verify  the  proposition  that  the  sum  of  the  moments 
about  any  axis,  of  forces  in  equilibrium^  is  zero. 

When  forces  act  on  an  extended  body,  they  are  not 
necessarily  concurrent.  The  condition  of  translational 
equilibrium,  however,  is  just  the  same  as  though  they 
were.  On  the  other  hand,  in  order  that  a  body  be  in 


56  THE  PRINCIPLE  OF  MOMENTS  [17 

rotational  equilibrium,  the  forces  must  also  have  zero, 
net,  rotating  effect.  The  rotating  effect,  or  moment  of 
a  force,  is  measured  by  the  product  of  the  force  by  the 
normal  distance  to  the  axis  about  which  rotation  is 
considered.  Since  a  body  in  equilibrium  is  in  rotation 
about  no  axis,  the  forces  must  have  zero  rotating  effect 
about  any  axis.  As  you  will  see  in  this  experiment, 
when  forces  have  zero  moment  about  one  axis,  their 
moment  about  any  other  axis  is  also  zero. 

The  apparatus  used  to  test  the  principle  consists  of  a 
circular  table  with  a  movable  disk  resting  on  bicycle  balls. 
The  disk  may  be  pivoted  in  the  center  if  desired.  To  pegs, 
placed  at  will  in  the  disk,  cords  are  attached  which  pass 
over  pulleys  clamped  at  different  points  around  the  cir- 
cular table.  From  the  ends  of  the  cords  are  suspended 
known  masses  whose  weight  produces  the  forces  required. 

(a)  Place  the  disk  on  four  bicycle  balls,  widely  sepa- 
rated, and  level  up  the  table  so  that  the  disk  will  not 
tend  to  move  in  any  one  direction  in  preference  to  another. 
Pivot  the  disk  in  the  center  and  place  a  sheet  of  manila 
paper   upon   it.      Attach   cords   to   it   at   three   different 
points  chosen  at  random;  and,  placing  the  pulleys  at  any 
convenient   points,   add   masses   until   two   of  the  forces 
have  large  values.     Adjust  the  third  force,   both  as  to 
magnitude    and    direction,    until    no    motion    results    on 
removing  the  peg.     See  that  the  disk  is  free  to   move 
on  the  bicycle  balls  and  that  the  cords  all  lie  in  a  plane 
close  to  and  parallel  to  the  top  of  the  disk;  then  mark 
points  or  lines  on  the  paper  to  indicate  the  directions  of 
the  forces.     Note  the  magnitude  of  each  force,  including 
the  weight  of  the  hanger. 

(b)  Repeat  (a),  using  four  forces  instead  of  three. 


17]  THE  PRINCIPLE  OF  MOMENTS  57 

(c)  On  the  papers  used,   trace  the  lines  of  direction 
of  the  forces,  and  make  the  necessary  measurements  to 
determine  their  moments  about  a  point  chosen  at  random. 
Find  the  sum  of  these  moments,  taking  those  as  positive 
which  tend  to  produce  a  counter-clockwise  rotation  about 
the   given   axis,    and   those   as    negative   which   tend   to 
produce  a  clockwise  rotation.     Select  in  turn  as  centers 
of  moments  three  points    as  widely  separated  as  possible. 
Find  the  sum  of  the  moments  for  each  of  these  centers 
and  do  this  for  both  (a)  and  (b). 

Put  your  work  in  neat  tabular  form.  Give  in  each 
case  the  percent  deviation  from  the  mean  between  the 
sum  of  the  positive  moments  and  the  sum  of  the  negative 
moments. 

(d)  Prolong  two  of  the  lines  in  (a)  until  they  inter- 
sect.     At    what    distance   from   the   intersection   is   the 
third  line  at  its  nearest  point?     Assuming  the  principle 
of  moments  to  be  true,  give  a  reason  why  three  forces 
in    equilibrium    must    be    concurrent.      Does    the    same 
reason  apply  to  four  forces?     Do  the  four  forces  in  (6) 
intersect  in  a  single  point? 

A  ladder  leaning  against  a  smooth  vertical  wall  is 
prevented  from  sliding  by  the  reaction  of  the  ground. 
Assuming  that  the  force  which  the  wall  exerts  against 
the  ladder  is  horizontal,  find,  by  construction,  the  force 
exerted  on  the  ladder  by  the  ground. 


58  THE  PARALLELOGRAM  LAW  [18 


18.    THE  PARALLELOGRAM  LAW:  STATICS 

To  verify  the  parallelogram  law  by  means  of  the  force- 
table. 

The  force-table  is  a  device  for  applying  known  forces 
at  a  common  point  so  as  to  make  known  angles  with  each 
other.  When  the  forces  are  in  equilibrium,  the  point  on 
which  they  act  will  not  change  position  when  released. 

The  law  will  be  verified  in  two  different  ways,  for  two 
different  force-groups. 

(1)  In   the   graphical   method,    the   forces   are   repre- 
sented by  lines.     Every  force  is  characterized  by  a  mag- 
nitude and  a  direction.     The  line  representing  the  force 
may  exhibit  the  magnitude  of  the  force  by  its  length, 
and  the  direction  by  its  direction.     It  should  be  drawn 
with  a  dot  at  one  end  and  an  arrowhead  at  the  other. 
A  diagram  in  which  each  force  is  represented  by  such 
a  line  is  called  a  force-diagram.     Unknown  forces  may 
be  determined  by  direct  measurement  on  such  a  diagram. 

(2)  In  the  method  of  components,  use  is  made  of  the 
proposition  that  when  a  point  is  in  equilibrium,  the  sum 
of  the  components  of  forces  acting  on  it  taken  parallel 
to  any  direction  is  zero. 

(a)  Set  up  the  force-table,  and  adjust  three  large 
forces,  not  equal,  and  not  forming  right-angles  with 
each  other.  Two  of  the  forces  should  be  set  with  the 
pin  in  place,  and  the  third  adjusted,  both  as  to  direction 
and  magnitude,  so  as  to  secure  equilibrium  when  the 
pin  is  removed.  Record  the  circular  scale  readings  and 
the  forces.  Include  the  weight  of  the  hangers  with  that 
of  the  weights. 


18]  THE  PARALLELOGRAM  LAW  59 

(b)     Repeat  (a),  using  four  forces  instead  of  three. 


(c)  Draw  the  force- diagram  for  each  of  the  two  cases 
accurately  to  scale,  with  the  protractor  and  dividers. 
In  the  first  case,  construct  a  parallelogram  on  any  two 
of  the  forces,  and  determine  their  equilibrant.  Record 
the  deviation  between  this  and  the  third  force,  both  as 
to  magnitude  and  direction.  In  the  second  case,  lay  off 
three  of  the  forces  in  succession  to  form  a  force-polygon, 
and  compare  the  vector  required  to  close  the  polygon 
with  the  fourth  force. 

Also  determine,  in  each  of  the  two  cases,  the  component 
of  each  of  the  forces  parallel  to  some  line  not  coinciding 
in  direction  with  any  of  the  forces.  No  diagram  is 
necessary,  but  you  may  find  it  convenient.  The  com- 
ponent is  calculated  simply  by  multiplying  the  magnitude 
of  the  force  by  the  cosine  of  the  angle  through  which  it 
is  projected.  To  estimate  error,  express  the  percent 
difference  from  the  mean  .of  the  algebraic  sums  of  the 
components. 

Tabulate  all  results  neatly. 

Solve  the  following  problem:  Three  men  stand  in  a 
row  on  top  of  a  wall  and  pull  a  heavy  load  up  the  wall 
by  means  of  ropes  attached  to  the  load.  The  angles 
at  which  the  ropes  act  on  the  load  are  +22°,  -12°,  and 
-25°,  measured  from  the  vertical.  Each  man  pulls  with 
the  same  force.  What  fraction  of  the  load  does  each 
man  lift?  Calling  the  load  W  and  the  force  each  man 
exerts  F,  write  the  equation  for  vertical  components. 
Solve  this  equation  for  F,  and  then  find  the  vertical 
component  of  the  force  exerted  by  each  man.  Check 
your  solution  by  adding  vertical  components  to  see  if 
they  equal  the  total  load. 


60  PHYSICAL  TABLES 

USEFUL  NUMERICAL  RELATIONS. 
Mensuration. 

Circle:  circumference  =  27rr;  area  =  7rrz. 
Sphere:  area  =  47Tr2;  volume  =  ^TTr3. 
Cylinder:  volume  =  7T>2/. 

Angle. 

1  radian  =  57°.2958  =  3437'.75. 
1  degree  =  0.017453  radian. 

Length. 

1  centimeter  (cm.)  =  0.3937  in.  1  inch  (in.)    =  2.540  cm. 

1  meter  (m.)  =  3.281  ft.  1  foot  (ft.)    =  0.3048  m. 

1  kilometer  (km.)  =  0.6214  mi.  1  mile  (mi.)  =  1.609  km. 

1  micron  (M)  =  0.001  mm.  1  mil  =  0.001  in. 

Area. 

1  sq,  cm.  =*  0.1550  sq.  in.  1  sq.  in.  =  6.451  sq.  cm. 

1  sq.  m.    -  10.674  sq.  ft.  1  sq.  ft.  =  0.09290  sq.  m. 

Volume. 

1  cc.  =  0.06103  cu.  in.  1  cu.  in  =  16.386  cc. 

1  cu.  in.  =  35.317  cu.  ft.  1  cu.  ft.  =  0.02832  cu.  m. 

1  liter  (1000  cc.)  =  1.7608  pints.  1  quart  =  1.1359  liters. 

Mass. 

1  gram  (gm.)          =  15.43  gr.  1  grain  (gr.)    =  0.06480  gm. 

1  kilogram  (kg.)    =  2.2046  Ib.          1  pound  (Ib.)  =  0.45359  kg. 

Density. 

1  gm.  per  cc.    =  62.425  Ib.  per  cu.  ft. 
1  Ib.  per  cu.  ft.  =  0.01602  gm.  per  cc. 

Thermometric  Scales. 

C=5  (F  — 32)/9  F  =  (9C/5)  4  32 

(C— centigrade  temperature;  F  =  Fahrenheit  temperature) 


PHYSICAL  TABLES  61 


USEFUL  NUMERICAL  RELATIONS. 

/  Force. 

1  gram's  weight  (gm.  wt.)  =     980.6  dynes  (g,  =  980.6  cm./sec.'.) 
1  pound's  weight  (Ib.  wt.)    =  0.4448  megadynes  (g0  =  980.6.) 

(The  "gm.  wt.'  is  here  defined  as  the  force  of  gravity  acting  on 
a  gram  of  matter  at  sea -level  and  45°  north  latitude.  The  "Ib.  wt."  is 
similarly  defined.) 

Pressure  and  Stress 

1  cm.  of  mercury  at  0°C.  1  in   of  mercury  at  0°C. 

=  13.596  gm.  wt.  per  sq.  cm.  =  34.533  gm.  wt.  per  sq.  cm. 

=  0.19338  Ib.  wt.  per  sq.  in.  =  0.49118  Ib.  wt.  per  sq.  in. 

Work  and  Energy. 

1  kilogram-meter  (kg.  m.)  =  7.233  ft.  Ib. 

1  foot-pound  (ft.  Ib.)  =  0.13826  kg.  m. 

1  joule  =  107  ergs. 

1  foot-pound  =  1.3557  X  107  ergs.  (g0=  980.6  cm./sec.s.) 

1  foot-pound  =  1.3557  joules  (g0  =  980.6.) 

1  joule  =  0.7376  ft.  Ib    (g0  =  980.6.) 

Power  (or  Activity). 

1  horse-power  (H.  P.)  =  33000  ft.  Ib.  per  min. 

1  watt  =  1  joule  per  sec.  =  107  ergs  per  sec. 

1  horse-power  =  745.64  watts  (g0  =  980.6  cm./sec.1) 

1  watt  =  44.28  ft.  Ib.  per  min  (g0  =  980.6) 

Mechanical  Equivalent. 

I  gro  -calorie  =  4.187  X  107  ergs. 

=  0.4269  kg.  m.  (g,  =  980.6  cm./sec.2.) 
=  3.088  ft.  Ib.  (g0  =  980.6.) 


62 


PHYSICAL  TABLES 


DENSITY  OF  DRY  AIR. 

(Values  are  given  in  gms.  per  cc.) 


Temp. 

Barometric  Pressure  (Centimeters  of  Mercury) 

C. 

72 

73 

74 

75 

76 

77 

0° 

.001225 

.001242 

.001259 

.001276 

.001293 

.001310 

1 

220 

237 

254 

271 

288 

305 

2 

216 

233 

250 

267 

283 

300 

3 

212 

228 

245 

262 

279 

296 

4 

207 

224 

241 

257 

274 

290 

5° 

.001203 

.001219 

.001236 

.001253 

.001270 

.001286 

6 

198 

215 

232 

248 

265 

282 

7 

194 

211 

227 

244 

260 

277 

8 

190 

206 

223 

239 

256 

272 

9 

186 

202 

219 

235 

251 

268 

10° 

.001181 

.001198 

.001214 

.001231 

.001247 

.001263 

11 

177 

194 

210 

226 

243 

259 

12 

173 

189 

206 

222 

238 

255 

13 

169 

185 

202 

218 

234 

250 

14 

165 

181 

197 

214 

230 

246 

15° 

.001161 

.001177 

.001193 

.001209 

.001225 

.001242 

16 

157 

173 

189 

205 

221 

237 

17 

153 

169 

185 

201 

217 

233 

18 

149 

165 

181 

197 

213 

229 

19 

145 

161 

177 

193 

209 

224 

20° 

.001141 

.001157 

.001173 

.001189 

.001204 

.001220 

21 

137 

153 

169 

185 

200 

216 

22 

133 

149 

165 

4 

181 

196 

212 

23 

130 

145 

161 

^177 

192 

208 

24 

126 

141 

157 

173 

188 

204 

25° 

.001122 

.001138 

.001153 

.001169 

.001184 

.001200 

26 

118 

134 

149 

165 

180 

196 

27 

114 

130 

145 

161 

176 

192 

28 

110 

126 

142 

157 

172 

188 

29 

107 

122 

138 

153 

169 

184 

30° 

.001103 

.001119 

.001134 

.001149 

.  01165 

.001180 

Corrections  for  Moisture  in  the 

Atmosphere 

Dew-point  Subtract  Dew-point 

Subtract 

Dew-point 

Subtract 

—10°    .000001     +  2° 

.000003 

-4-14° 

.000007 

—  8         2     +4 

4 

+  16 

8 

—  6         2     +6 

4 

+  18 

9 

—  4         2     +8 

5 

+  20 

.000010 

—  2         3     +10 

6 

+  24 

13 

0         3     +12 

6 

+  28 

16 

PHYSICAL  TABLES 


63 


DENSITIES    AND    THERMAL    PROPERTIES    OF    GASES. 

(The  densities  are  given  at  0°C.  and  76  cm.  pressure,  and  the 
specific  heats  at  ordinary  temperatures.  The  coefficients  of  cubical 
expansion  (at  constant  pressure)  of  the  gases  listed  below  are  not 
given  in  this  Table;  they  are  about  the  same  for  all  the  permanent 
gases,  being  approximately  1/273  or  0.003663,  if  referred  in  each 
case  to  the  volume  of  the  gas  at  0°C.  The  specific  heats  at  constant 
pressure  and  at  constant  volume  are  represented  by  the  symbols 
Sp.  and  Sv) . 


Gas  or  Vapor 

Formula 

Density 
(gms.  per  cc.) 

Molecular 
Weight 

Sp  ~5~  Sv 

(cafs. 
per  gm.) 

Air 

n  nm  OQQ 

1/11 

OOQ7 

Ammonia 
Carbon  dioxide 
Carbon  monoxide 
Chlorine 
Hydrochloric  acid 
Hydrogen 
Hydrogen  sulphide 
Nitrogen,  pure 

NH3 
C02 
CO 
Cli 

HC1 
H2 
H2S 

N2 

0.000770 
0.001974 
0.001234 
0.003133 
0.001616 
0.0000896 
0.001476 
0.001254 
n  001  9^7 

17.06 
44.00 
28.00 
70.90 
36.46 
2.016 
34.08 
28.08 

.41 

1.33 
1.29 
1.40 
1.32 
1.40 
.41 
.34 
.41 

.Zot 

.530 
.203 
.243 
.124 
.194 
3.410 
.245 
.244 

Oxygen 
Steam  (100°C.) 
Sulphur  dioxide 

02 
H2O 

S02 

0.001430 
0.000581 
0.002785 

32.00 
18.02 
64.06 

.41 
.28 
.26 

.218 
.421 
.154 

DENSITY  AND  SPECIFIC  VOLUME  OF  WATER. 


Temp. 
C. 

Density 
(gms.  per  cc.) 

Specific 
Volume 
(cc.  per  gm.) 

Temp. 
C. 

Density 
(gms.  per  cc.) 

Specific 
Volume 

(cc.  per  gm.) 

0 

0.999868 

1.000132 

20° 

0.99823 

1.00177 

1 

927 

073 

25 

777 

294 

2 

968 

032 

30 

567 

435 

3 

992 

008 

35 

406 

598 

3.98 

1.000000 

000 

40 

224 

782 

5 

.999992 

008 

50 

.98807 

1.01207 

6 

968 

032 

60 

324 

705 

7 

929 

071 

70 

.97781 

1.02270 

8 

876 

124 

80 

183 

902 

9 

808 

192 

90 

.96534 

1.03590 

10 

727 

273 

100 

.95838 

1.04343 

15 

126 

874 

102 

693 

501 

64 


PHYSICAL  TABLES 


DENSITIES   AND   THERMAL   PROPERTIES   OF   LIQUIDS. 

(The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  liquids  listed.  The  student  should  not  expect  the  properties 
of  the  average  laboratory  specimen  to  correspond  exactly  in  value 
with  them.  With  a  few  exceptions  the  densities  are  given  for  ordi- 
nary atmospheric  temperature  and  pressure.  The  specific  heats  and 
coefficients  of  expansion  are  in  most  cases  the  average  values  be- 
tween 0°  and  100°C.  The  boiling  points  are  given  for  atmospheric 
pressure,  and  the  heats  of  vaporization  are  given  at  these  boiling 
points.) 


* 

o 

ill 

ba 

c 
o 

»- 

MM 

•5     ~£rt 

*7|   'Q     co 

C    •£ 

O    rt 

C 

W     o> 

Jfi    ^    d 

^  "o 

+-1  .£i 

Liquid 

Q 

£K 

o?i 

«A< 

d    >- 

(LI     O 

(calories 

> 

per  gm. 

(per  degree 

(degrees 

(calories 

(gms.  per  cc.) 

per  deg.) 

C.) 

C.) 

per  gm.) 

Alcohol  (ethyl) 

0.794 

.58 

.00111 

78 

205* 

Alcohol  (methyl) 

.796 

.60 

.00143 

66 

262f 

Benzene 

.880 

.42 

.00123 

80 

93.2 

Carbon  bisulphide 

1.29 

.24 

.00120 

466 

84 

Cotton  seed  oil 

.925 

.47 

.00077 

Ether 

.74  (0°C) 

.55 

.00162 

35 

90 

Glycerine 

1.26 

.576 

.000534 

Hydrochloric  acid 

1.27 

.75 

.000455 

110 

Mercury 

13.596  (0°) 

.033 

.0001815 

357 

67 

Olive  oil 

.918 

.47 

.000721 

Nitric  acid 

1.56 

.66 

.00125 

86 

115 

Sea-water 

1.025 

.938 

Sulphuric  acid 

1.85 

.33 

.00056 

338 

122 

Turpentine 

.873 

.47 

.00105 

159 

70 

*  The  heat  of  vaporization  of  ethyl  alcohol  at  0°C.  is  236.5. 
t  The  heat  of  vaporization  of  methyl  alcohol  at  0°C.  is  289.2. 


PHYSICAL  TABLES 


65 


DENSITIES    AND    THERMAL    PROPERTIES    OF    SOLIDS. 

(The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  substances  listed.  The  student  should  not  expect  the  prop- 
erties of  the  average  laboratory  specimen  to  correspond  exactly 
in  value  with  them.  As  a  rule  the  densities  are  given  for  ordi- 
nary atmospheric  temperature.  The  specific  heats  and  coefficients 
of  expansion  are  in  most  cases  the  average  values  between  0°  and 
100°  C.  The  melting  points  and  heats  of  fusion  are  given  for 
atmospheric  pressure.) 

The  coefficient  of  cubical  expansion  of  solids  is  approximately 
three  times   the   linear  coefficient. 


*j   U    fl 

^v 

o 

S  rt  o 

C)    bfl 

<+-  1 

+J 

«!+• 

•-  2  '55 

O   g 

Ti  d 

o  c  £ 

g 

CJ    a> 

£K 

*§3  a 

§|2 

2  3 

Solid. 

Q 

CO 

CJ^W 

ffi^ 

cals.  per 

degrees 

cals.  per 

gms.  per  cc. 

gm. 

per  degree  C. 

C. 

gm. 

Acetamide                       1.56 

82 

Aluminum 

2.70 

0.219 

.0000231 

658 

Brass,  cast 

8.44 

.092 

.0000188 

"       drawn 

8.70 

.092 

.0000193 

Copper 

8.92 

.094 

.0000172 

1090 

43.0 

German-silver 

8.62 

.0946 

.000018 

860 

Glass,  common  tube 

2.46 

.186 

.0000086 

"  .    flint 

3.9 

.117 

.0000079 

Gold 

19.3 

.0316 

.0000144 

1065 

Hyposul.  of  soda 

1.73 

.445 

48 

Ice 

.918 

.502 

.000051 

0 

80. 

Iron,  cast 

7.4 

.113      i.  0000106 

1100 

23-33 

"     wrought 

7.8 

.115       .000012 

1600 

Lead 

11.3 

.0315     .000029 

326 

5.4 

Mercury 

13.596 

.0319 

—39 

2.8 

Nickel 

8.90 

.109 

.0000128       '  1480 

4.6 

Paraffin,  wax 

.90 

.560 

.000008-23  i      52 

35.1 

liquid 

.710 

Platinum 

21.50 

.0324 

.0000090 

1760 

27.2 

Rubber,  hard 

1.22 

.331 

.000064 

Silver 

10.53 

.056 

.0000193 

960 

21.1 

Sodium  chloride 

2.17 

.214 

.000040 

800 

Steel 

7.8 

.118 

.000011 

1375 

Wood's  alloy,  solid 

9.78 

.0352 

75.5 

8.40 

"     ,  liquid 

.0426 

66 


PHYSICAL  TABLES 


VISCOSITY  OF  AQUEOUS  SOLUTIONS  OF  SUGAR. 


%  Sugar 

Coeff.  at  20°  C. 
(C.  G.  S.  Units.) 

Coeff.  at  30°C. 
(C.  G.  S.  Units.) 

0 
5 
10 
20 
40 

0.0100 
.0117 
.0132 
.0191 
.0600 

0.0080 
.0089 
.0104 
.0145 
.0423 

COEFFICIENTS  OF  FRICTION. 


Substances. 

Static  Coefficient. 

Kinetic  Coefficient 

Metals  on  metals  (dry) 
"       (wet) 
(oiled) 
Wood  on  wood  (dry) 
(a)  direction  of  fiber 
(b)  normal  to  fiber 
Leather  belt  on  wood  pulley 
"     "     iron      " 

fro 

m  0.2    1 
0.15 
0.15 

0.5 
0.4 
0.45 
0.25 

:o  0.4 
0.3 
0.2 

0.7 
0.6 
0.6 
0.35 

fro 

m  0.18  1 
0.14 
0.14 

0.2 
0.18 
0.3 
0.2 

.0  0.35 
0.28 
0.18 

0.3 
0.3 
0.5 
0.3 

ELASTIC  CONSTANTS  OF  SOLIDS. 
(Approximate  Values.) 


Substance 

Bulk-Modulus. 
(C.G.S.  Units.) 

Simple  Rigidity 
(C.G.S.Units.) 

Young's  Modulus 
(C.G.S.Units.) 

Aluminum 
Brass,  drawn 
Copper 

5.5  x  1011 
10.8  x    " 
16.8  x    " 

2.5  x  1011 
3.7  x 
4.5  x 

4K   Y 

6.5  x  10n 
10.8  x 
12.3  x 

10  Q   Y 

Place 

24.    Y 

7  0  * 

Iron,  wrought 
Steel 

14.6  x    " 
18.4  x    " 

7.7  x 
8.2  x 

19.6  x 
21.4  x 

PHYSICAL  TABLES 


67 


SURFACE  TENSION  OF  PURE  WATER  IN  CONTACT 
WITH  AIR. 

(The  value  of  the  surface  tension  of  a  liquid  is  dependent  only 
upon  the  character  and  temperature  of  the  liquid  and  upon  the 
nature  of  the  gas  above  the  surface  of  the  liquid.  It  is  independent  of 
the  curvature  of  the  surface  film  and  of  the  material  of  the  con- 
taining vessel.) 


Temp. 
C. 

Tension 
(dynes  pr.  cm) 

Temp. 
C. 

Tension 
(dynes  pr.  cm) 

Temp. 

c. 

Tension 
(dynes  pr.cm) 

0° 
5 
10 
15 
20 
25 

75.5 

74.8 
74.0 
73.3 
72.5 

71.8 

30° 
35 
40 
45 
50 
55 

71.0 
70.3 
69.5 
68.6 
67.8 
66.9 

60° 
65 
70 
80 
100 

crit.  temp 

66.0 
65.1 
64.2 
62.3 
56.0 
0.0 

SURFACE   TENSIONS   OF   SOME   LIQUIDS   IN   CONTACT 
WITH  AIR. 

(The  angle  of  contact  between  pure  water  and  clean  glass  vessels 
of  all  sizes  is  0°;  the  angle  of  contact  between  pure  water  and  clean 
steel  or  silver  is  about  90°;  the  angle  of  contact  between  mercury 
and  glass  is  about  132°.  See  the  note  to  Table  VIII.) 


Dynes 
per  cm. 

Dynes 
per  cm. 

Alcohol  (ethyl)      at  20° 
Alcohol  (methyl)  at  20° 
Benzene                   at  15° 
Glycerine                at  18° 

22-24 
22-24 
28-30 
63-65 

Mercury         at  20° 
Olive  oil          at  20° 
Petroleum      at  20° 
Water  (pure)  at  20° 

470-500 
32-36 
24-  26 

72-  74 

VISCOSITY  OF  WATER. 


Temp. 
C. 

Coeff.  of  Vise. 
(C.G.S.Units) 

Temp. 
C. 

Coeff.  of  Vise. 
(C.G.S.Units) 

Temp. 
C. 

Coeff.  of  Vise. 
(C.G.S.Units) 

0° 
5 
10 
15 
20 

0.0178 
.0151 
.0131 
.0113 
.0100 

25° 
30 
35 
40 
50 

0.0089 
.0080 
.0072 
.0066 
.0055 

60° 
70 
80 
90 
100 

0.0047 
.0041 
.0036 
.0032 
.0028 

68 


PHYSICAL  TABLES 


(a)  BOILING  POINT  OF  WATER  AT  DIFFERENT  BARO- 

METRIC PRESSURES. 

(b)  VAPOR-PRESSURE  OF  SATURATED  WATER-VAPOR. 

(This  table  may  be  used  either  (a)  to  find  the  boiling  point  /  of 
water  under  the  barometric  pressure  P,  or  (b)  to  find  the  vapor- 
pressure  P  of  water-vapor  saturated  at  the  temperature  /,  the  dew- 
point.) 


t° 
c. 

P 
cm. 

D 

gm./cc. 

t° 
C. 

P 
cm. 

D 
gm./cc. 

t° 
C. 

P 
cm. 

D 
gm.  /cc. 

-10° 

.22 

2.3xlO-6 

30° 

3.15 

30.1x10-° 

88.5  49.62 

-  9 

.23 

2.5x  " 

35  !  4.18 

39.3x" 

89 

50.58 

-  8 

.25 

2.7x" 

40   5.49 

50.9x" 

89.5 

51.55 

-  7 

.27 

2.9x" 

45   7.14 

65.3x  " 

90 

52.54 

428.4xlO-6 

-  6 

.29 

3.2x  " 

50 

9.20 

83.0x  " 

90.5 

53.55 

-  5 

.32 

3.4x  " 

55 

11.75 

104.6x  " 

91 

54.57 

-  4 

.34 

3.7x  " 

60   14.88 

130.7x  " 

91.5 

55.61 

-  3 

.37 

4.0x  " 

65 

18.70 

162.1x" 

92 

56.67 

-  2 

.39 

4.2x  " 

70 

23.31 

199.5x  " 

92.5 

57.74 

-  1 

.42 

4.5x  " 

71 

24.36 

93 

58.83 

0 

.46 

4.9x  " 

72 

25.43 

93.5 

59.96 

1 

.49 

5.2x  " 

73 

26.54 

94 

61.06 

2 

.53 

5.6x  " 

74 

27.69 

94.5 

62.20 

3 

.57 

6.0x  " 

75 

28.88 

243.7  " 

95 

63.36 

511.1" 

4 

.61 

6.4x  " 

75.5  |29.49 

95.5 

64.54 

5 

.65 

6.8x  " 

76  130.11 

96 

65.74 

6 

.70 

7.3x  " 

76.5  J30.74 

96.5 

66.95 

7 

.75 

7.7x  " 

77 

31.38 

97 

68.18 

8 

.80 

8.2x  " 

77.5  32.04 

97.5 

69.42 

9 

.85 

8.7x  " 

78  132.71 

98 

70.71 

10 

.91 

9.3x  " 

78.5133.38 

98.2 

71.23 

11 

.98 

lO.Ox  " 

79   34.07 

98.4 

71.74 

12 

1.04 

10.6x  " 

79.5  34.77 

98.6 

72.26 

13 

1.11 

11.2x" 

80   35.49 

295.9  " 

98.8 

72.79 

14 

1.19 

12.0x  " 

80.5  :36.21 

99 

73.32 

15 

.27 

12.8x  " 

81   36.95 

99.2 

73.85 

16 

.35 

13.5x  " 

81.5  37.70 

99.4 

74.38 

17 

.44 

14.4x  " 

82   38.46 

99.6J  74.92 

18 

.53 

15.2x  " 

82.5  39.24 

99.8 

75.47 

19 

.63 

16.2x  " 

83  J40.03 

100 

76.00 

606.2  " 

20 

.74 

17.2x  " 

83.5  '40.83 

100.2 

76.55 

21 

.85 

18.2x  " 

84 

41.65 

100.4 

77.10 

22 

1.96 

19.3x  " 

84.5 

42.47 

100.6 

77.65 

23 

2.09 

20.4x  " 

85 

43.32 

357.1  " 

100.8 

78.21 

24 

2.22 

21.6x" 

85.5144.17 

101 

78.77 

25 

2.35i22.9x  " 

86   45.05 

102 

81.60 

26 

2.50|24.2x  " 

86.5i45.93 

103 

84.53 

27 

2.65 

25.6x  " 

87   46.83 

105 

90.64 

715.4  " 

28 

2.81 

27.0x  " 

87.5  J47.74 

107 

97.11 

29 

2.9728.5x" 

88  :48.68 

110 

107.54 

840.1  " 

PHYSICAL  TABLES 


69 


THE  WET-  AND  DRY-  BULB  HYGROMETER.  DEW-POINT. 

(This  Table  gives  the  vapor-pressure,  in  mercurial  centimeters,  of 
the  water-vapor  in  the  atmosphere  corresponding  to  the  dry-bulb 
reading  /°C.  (first  column)  and  the  difference  (first  row)  between 
the  dry-bulb  and  wet-bulb  readings  of  the  hygrometer.  Having 
obtained  from  this  Table  the  value  of  the  vapor-pressure  for  a  given 
case,  the  dew-point  can  be  found  by  consulting  Table  XIV.  The  data 
given  below  are  calculated  for  a  barometric  pressure  equal  to  76  cm.) 


jo/^j          Difference  between  Dry-bulb  and  Wet-bulb  Readings. 

0°       1° 

Oo 

3'       4° 

5° 

6° 

7°        8°       9°      10° 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

10° 

.92 

.81 

.70 

.60 

.50 

.40 

.31 

.22 

.13 

11 

.98 

.87 

.76 

.65 

.55 

.45 

.35 

.26 

.17 

12 

.105 

.93 

.82 

.71 

.60 

.50 

.40 

.30 

.21 

.12 

.03 

13 

1.12 

.00 

.89 

.76 

.65 

.55 

.45 

.35 

.25 

.16 

.07 

14 

1.19      .07 

.94 

.83 

71 

.61 

.50 

.40 

.30 

.20 

.11 

15 

1.27!     .14 

.01 

.90 

.78 

.66 

.55 

.45      .34 

.25 

.15 

16 

1.35)     .22 

.09 

.97 

.84 

.73 

.60 

.50      .40 

.30 

.19 

17 

1.44 

.30 

.17    1.04 

.91 

.80 

.67 

.56 

.45 

.35 

.24 

18 

1.54 

.39 

.25 

1.12 

.99 

.86 

.74 

.63      .51 

.40 

.30 

19 

1.63 

.49 

.34 

1.20    1.07 

.94 

.81 

.69      .57      .46 

.35 

20 

1.74 

.59 

.43 

1.29    1.15 

1.02 

.88 

.76      .64      .52 

.41 

21 

1.85 

.69 

.53 

1.38    1.24 

1.10 

.96 

.84      .71|     .59 

.47 

22 

1.97 

.80 

.64 

1.48    1.33 

1.19 

1.05 

.91 

.78!     .66 

.54 

23 

2.09 

1.92 

.75 

1.59    1.43 

1.28 

1.13 

.00      .86;     .73 

.61 

24 

2.22 

2.04 

.86 

1.70    1.53 

1.38 

1.23 

.09      .94|     .81 

.68 

25 

2.35    2.17 

.99 

1.81    1.64 

1.48 

1.33 

.18    1.03J     .90 

.76 

26 

2.50 

2.31 

2.11 

1.94    1.76 

1.59 

1.43 

.28    1.13      .98      .84 

27 

2.65    2.45 

2.25 

2.07    1.88 

1.71 

1.54 

.38    1.23    1.08 

.93 

28 

2.81J  2.60 

2.40 

2.20    2.01 

1.83 

1.66 

.49    1.33    1.18 

1.02 

29 

2.98    2.76 

2.55 

2.35    2.15 

1.96 

1.78 

.61    1.44    1.28 

1.12 

30 

3.15    2.93 

2  71 

2.50    2.29 

2.10 

1.91 

.73    1.55    1.39    1.23 

Miscellaneous. 

(1.)      Heat  of  Neutralization. 

Any  strong  acid  with  any  strong  alkali  evolves   (+)  about 
761  calories  for  every  gm.  of  water  formed. 

(2.)      Heat  of  Solution  in  water. 

For  Calcium  oxide  (CaO),  -f  327  cals.  per  gm. 

"    Sodium  chloride  (NaCl),  -     21     "       "     " 

hydroxide  (NaOH),  +  248     "       "     " 

hyposulphite  (Na2S2O3+5H2O),  -    44.8 "       "     " 

(3.)     Fuel  value  of  illuminating  gas  is  5500  to  6500  calories  per  liter, 
its  density  is  .00058  gm.  per  cc.  at  0°C  and  76  cm.  pressure. 

Fuel  value   of  ethyl  alcohol    is  7400,  of  methyl    alcohol    5700, 
calories  per  gram. 


70 


NATURAL  SINES. 


0' 

6 

12 

18 

24' 

30' 

36 

42' 

48' 

54' 

123 

4  5 

0° 

0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

3  6  9 

12  15 

1 
2 
3 

0175 
0349 
0523 

0192 
0366 
0541 

0209  0227 
0384  0401 
0558  0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
0663 

0332 
0506 
0680 

3  6  9 
3  6  9 
3  6  9 

12  15 
12  15 

12  15 

4 
5 
6 

0698 
0872 
1045 

0715107320750 
0889  0906  092^ 
1063!  1080  1097 

0767 
0941 
IH5 

07«5 
0958 
1132 

0802 
0976 
1149 

O8ig 

°993 
1167 

0837 
ion 
1184 

0854 
1028 

1201 

3  6  9 
369 
3  6  9 

12  15 
12  14 
12  14 

7 
8 
9 

1219 
1392 
1564 

1236 
1409 
1582 

1253  1271 
1426  1444 
I599ji6i6 

1288 
1461 
1633 

1305 
1478 
1650 

i323 
1495 
1668 

1340 

1513 
1685 

1357 
1530 
1702 

1374 
1547 
1719 

3  6  9 
3  6  9 
3  6  9 

12  14 
12  14 

12  14 

10 

1736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

369 

12  14 

11 
12 
13 

1908 
2079 
2250 

£925 
2096 
2267 

1942 
2113 
2284 

'959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

201  1 

2181 

2351 

2028 
2198 
2368 

2045 
2215 

2385 

2062 
2232 
24O2 

369 
369 
368 

II  14 
II  14 
II  14 

14 
15 
16 

24*19 

2588 
2756 

2436 
2605 
2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 

2857 

2538 
2706 
2874 

2554 
2723 
2890 

2571 
2740 
2907 

368 
368 
368 

II  14 
II  14 
II  14 

17 
18 
19 

2924 
3090 
3256 

2940 
3107 
3272 

3437 

2957 
3123 
3289 

3453 

2974 
3140 
3305 

2990 
3156 
3322 
3486 

3007 
3173 
3338 

3O2^ 
3190 

3355 

3040 
3206 
3371 

3057 
3223 
3387 

3074 
3239 
3404 

368 
368 

3  5  8 

II  14 
II  14 
II  14 

20 

3420 

3469 

3502 

35i8 

3535 

355i 

3567 

3  5  8 

II  14 

21 
22 
23 

3584 
3746 
39°7 

3600 
3762 
3923 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 
3971 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
^019 

3714 
3875 
4035 

3730 
3891 
4051 

3  5  8 
3  5  8 
3  5  8 

II  14 
II  14 
II  14 

24 
25 
26 

4067 
4226 
4384 

4083 
4242 
4399 

4099 
4258 
4415 

4"5 
4274 

4431 

4131 
4289 
4446 

4M7 
4305 
4462 

4163 
4321 

4478 

4179 
4337 
4493 

4195 
4352 
4509 

4210 

4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II  13 
II  13 

10  13 

27 
28 

29 

4540 
4695 

4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
474i 
4894 

4602 
4756 
1909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
4970 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 
10  13 

30 

5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

3  5  8 

10  13 

31 
32 
33 

5i5o 
5299 
5446 

5165 
53M 
546i 

5180 
5329 
5476 

5195 
5344 
5490 

5210 

5358 
5505 

5225 
5373 
5519 

5240 

5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 

5577 

257 

2  5  7 
2  5  7 

10  12 
IO  12 
10  12 

34 
35 
36 

5592 
5736 
5878 

5606 
5750 
5892 

5621 
5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

5707 
5850 

599° 

5721 

5864 
6004 

257 
2  5  7 
2  5  7 

IO  12 
10  12 

9  12 

37 
38 
39 

6018 

6i57 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 
6347 

6088 
6225 
6361 

6101 
6239 
6374 

6115 
6252 
6388 

6129 
6266 
6401 

6143 
6280 
6414 

257 
257 
247 

9  12 

9  " 
9  ii 

40 

6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

247 

9  ii 

41 

42 
43 

6561 
6691 
6820 

6574 
6704 

6833 

6587 
6717 

6845 

6600 
6730 

6858 

6613 
6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

6652 
6782 
6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
246 
246 

9  ii 
9  ii 
8  ii 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

034 

7046 

7059 

246 

8  10 

NATURAL  SINES. 


71 


0' 

6' 

7083 

12' 

7096 

18 

24 

30 

36 

7M5 

42' 

7157 

48' 

7169 

54 

7181 

123 

4      5 

45° 

7071 

7108 

7120 

7133 

246 

8     10 

46 
47 
48 

7193 
73M 
743i 

7206 
7325 
7443 

7218 
7337 

7455 

7230 
7349 
7466 

7242 
736i 
7478 

7254 
7373 
7490 

7266 
7385 
75oi 

7278 
7396 
7513 

7290 
7408 
7524 

7302 
7420 
7536 

246 
246 
246 

8     10 
8     10 
8     10 

49 
50 
51 

7547 
7660 

777i 

7558 
7672 
7782 

7570 
7683 
7793 

758i 
769-4 
7804 

7593 
7705 
7815 

7604 
7716 
7826 

7615 
7727 
7837 

7627 
7738 
7848 

7638 
7749 
7859 

7649 
7760 
7869 

246 
246 
245 

8      9 
7      9 
7      9 

52 
53 
54 

7880 
7986 
8090 

7891 

7997 
8100 

7902 
8007 
8m 

7912 
8018 
8121 

7923 
8028 
8131 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

79^5 
8070 
8171 

7976 
8080 
8181 

245 
2    3    5 
235 

7      9 
7      9 

7      8 

55 

8192 

8202 

8211 

8221 

8231 

$241 

8339 
8434 
8526 

8251 

8261 

8271 

8281 

2    3     5 

7      8 

56 
57 
58 

8290 

8387 
8480 

8572 
8660 
8746 

&300 
8396 
8490 

8310 
8406 
8499 

8320 
8415 
8508 

8329 
8425 
8517 

8348 
8443 
8536 

8358 
8453 
S_545 
8634 
8721 
8805 

8368 
8462 
8554 

8377 
8471 
8563 

235 
2    3    5 
2    3    5 

6      8 
6      8 
6      8 

59 
60 
61 

S58i 
8669 
8755 
8838 
8918 
8996 

8590 
8678 
8763 

8599 
8686 

8771 

8607 
8695 
8780 

8616 
8704 
8788 

8625 
8712 
8796 

8643 
8729 
8813 

8652 
8738 
8821 

3    4 
3    4 
3    4 

6      7 
6      7 
6      7 

62 

63 
64 

8829 
8910 

8988 

8846 
8926 
9°Q3 

8854 
8934 
9011 

8862 
8942 
9018 

8870 
8949 
9026 

8878 
8957 
9033 

8886 
8965 
9041 

8894 
8973 
9048 

8902 
8980 
9056 

3    4 
3    4 
3    4 

5      7 
5      6 
5      6 

65 

9063 

9070 

9078 

9150 
9219 
9285 

9085 

9092 
9164 
9232 
9298 

9100 
9171 
9239 
9304 

9107 
9178 
9245 
93U 

9114 
9184 
9252 
9317 

9121 

9128 

2    4 

5      ^ 

66 
67 
68 
69" 
70 
71 

9135 
9205 
9272 

9M3 
9212 
9278 

9157 
9225 
9291 

9191 
9259 
9323 

gigh 
9265 
9330 

2    3 
2     3 
2    3 

5      6 
4      6 
4      5 

9336 
9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
9415 
9472 

9361 
9421 
9478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 

9391 
9449 
9505 

2    3 
2    3 
2    3 

4      5 
4      5 
4      5 

9500 

72 
73 
74 
~75~ 

95U 
9563 
9613 

95!6 
9568 
b6i7 

9521 
9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9548 
9598 
9646 

9553 
9603 
9650 

9558 
9608 
9655 
9699 

2    3 

2      2 
2      2 

4      4 
3      4 
3      4 

9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

I      2 

3      4 

76 
77 
78 

9703 
9744 
9781 

9707 
9748 
9785 

9711 
9751 
9789 

9715 
9755 
9792 

9720 
9759 
9796 
9829 
9860 
9888 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9736 
9774 
9810 

9740 
9778 
9813 

2 
2 
2 

3      3 
3      3 
2      3 

79 
80 
81 

9816 
9848 
9877 

9820 

9851 
9880 

9823 
9854 
9882 

9826 

9857 
9885 

9833 
9863 
9890 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I             2 
0 
0 

2      3 

2        2 
2        2 

82 
83 
84 
85 

9903 
9925 
9945 

9905 
9928 

9947 

9907 
9930 
9949 
9965 

9910 
9932 
995i 
9966 

9912 
9934 
9952 

9914 
9936 
9954 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

O 
0 
0 

2 
2 
I 

9962 
9976 
9986 
9994 

9963 

9968 

9969 

9971 

9972 

9973 

9974 

O     O      I 

I 

86 
87 
88 

9977 
9987 
9995 

9999 

9978 
9988 
9995 

9979 
9989 
9996 

9980 
9990 
9996 

9981 
9990 
9997 

9982 
999  i 
9997 

9983 
9992 
9997 

9984 
9993 
9998 

9985 
9993 
9998 

O    O      I 
0    0     O 
0    O     0 

I 

I 
0        0 

89 

9998 

9999 

9999 

9999 

1.000 
nearly 

1.  000 
nearly 

1.000 

nearlv 

1.  000 

nearly 

I.OOO 
nearly 

o   o    a 

0        0 

72 


NATURAL  TANGENTS. 


0 

6 

12' 

18 

24' 

30' 

36 

42 

48 

54' 

123 

4  5 

0° 

1 

2 
3 

.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

369 

12  14- 

.0175 

•0349 
.0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
0577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

0314 
0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12  15 
12  15 
12  15 

4 
5 
6 

.0699 

.0875 
.1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
"39 

0805 
0981 
H57 

0822 
0998 
"75 

0840 
1016 
1192 

0857 
1033 

1210 

369 

369 
369 

12  15 
12  15 
12  I5 

7 
8 
9 

.1228 
.1405 

.1584 

1246 

1423 
1602 

1263 
1441 
1620 

1281 

1459 
1638 

1299 
1477 
1655 

1317 
1495 
1673 

1334 
1512 
1691 

1352 
1530 
1709 

1370 
1548 
1727 

1388 
I566 
1745 

369 
369 
369 

12  I5 
12  15 
12  I5 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

369 

12  15 

11 
12 
13 

.1944 

.2126 

.2309 

1962 
2144 
2327 
2512 
2698 
2886 

1980 
2162 

2345 

1998 
2180 
2364 

2016 
2199 

2382 

2035 
2217 
2401 

2053 
2235 
2419 

2071 
2254 
2438 

2089 
2272 
2456 

2107 
2290 
2475 
2661 
2849 
3038 

369 

3     6     9 
369 

12  15 

12  15 
12  15 

14 
15 
16 

•2493 
.2679 
.2867 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 

2623 
28ll 
3000 

2642 
2830 
3019 

369 
369 
3     6     9 

12  l6 

13  16 
13  16 

17 
18 
19 

.3057 
•3249 

•3443 

3076 
3269 
3463 

3096 

3288 
3482 

3"5 
3307 
3502 

3134 
3327 
3522 

3153 
3346 
3541 

3172 
3365 
356i 

3191 

3385 
3581 

3211 
3404 
3600 

3230 
3424 
3620 

3     6    10 
3     6    10 
3     6    10 

13  16 
13  16 
13  17 

20 

•3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3      7    '0 

'3  17 

21 
22 
23 

•3839 
.4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

4307 

3919 
4122 

4327 

3939 
4142 

4348 

3959 
4163 
4369 

3978 
4183 
4390 

4000 
4204 
44" 

4020 
4224 
4431 

3      7    10 
3     7    '0 
3     7    10 

13  17 

»4  '7 
14  17 

24 
25 
26 

•4452 
.4663 

.4877 

4473 
4684 
4899 

4494 
4706 
4921 

4515 
4727 
4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 
479i 
5008 

4599 
4813 
5029 

4621 
4834 
5051 

4642 
4856 
5073 

4      7    10 
4      7    ii 
4      7    ii 

14  18 
14  18 
15  18 

27 
28 
29 

•5095 
•5317 
•5543 

5H7 
5340 
5566 

5139 
5362 
5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

5250 
5475 
5704 

5272 
5498 
5727 

5295 
5520 
5750 
5985 

4      7    M 
8    ii 

8      12 

15  18 
"5  19 
'5  19 

30 

•5774 
.6009 
.6249 
.6494 

5797 

58205844 

5867 

5890 

59M 

5938 

596.1 

8      12 

l6  20 

31 
32 
33 

6032 
6273 
6519 

6056 
6297 
6544 

6080 
6322 
6569 

6lO4 
6346 
6594 

6128 

6371 
6619 

6152 

6395 
6644 

6176 
6420 
6669 

6200 
6445 
6694 

6224 
6469 
6720 

8      12 
8      12 

S    13 

1  6  20 

16  20 

17  21 

34 
35 
36 

6745 
.7002 
•7265 

6771 
7028 
7292 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 

7373 

6873 
7133 
7400 

6899 
7159 
7427 

6924 
7186 
7454 

6950 
7212 
7481 

6976 

7239 
7508 

9    '3 
9    '3 
5     9    "4 

17  21 

l8  22 
l8  23 

37 
38 
39 

•7536 
•7813 
.8098 

7563 
7841 
8127 

7590 
7869 
8156 

7618 
7898 
8185 

7646 
7926 
8214 

7673 
7954 
8243 

7701 

7983 
8273 

7729 
8012 
8302 

7757 
8040 
8332 

7785 
8069 
8361 

5      9    '4 
5    10    '4 
5    i°    '5 

18  23 
19  24 

20  24 

40 

.8391 

8421 

845i 

8481 

8511 

8541 

8571 

8601 
8910 
9228 
9556 

8632 

8662 

5    '0    15 

20  25 

41 
42 
43 

.8693 
.9004 
•9325 

8724 
9036 
9358 

8754 
9067 

9391 

8785 
9099 

9424 

8816 
9131 
9457 

8847 
9163 
9490 

8878 
9195 
9523 

8941 
9260 
9590 

8972 
9293 
9623 

5    10    16 
5    ii    16 
6    ii    17 

21  26 
21  27 
22  28 

44 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

6    ii    17 

23  29 

NATURAL  TANGENTS. 


73 


45° 

0 

6' 

12 

18 

24' 

30 

36 

42' 

48 

54 

123 

4    5 

24  30 

1.  0000 

0035 

0070 

0105 

0141 

0176 

O2I2 

057? 
0951 
1343 

0247 

0283 

0319 

6    12    18 

46 
47 
48 

1-0355 
1.0724 
1.1106 

0392 
0761 
H45 

0428 
0799 
1184 

0464 
0837 
1224 

0501 
0875 
1263 

0538 
0913 
1303 

0612 
0990 
383 

0649 
1028 
1423 

0686 
1067 
1463 

6    12    18 
6   43    19 
7    13    20 

25    3' 
25    32 
26     33 

49 
50 
51 
52 
53 
54 
"55 

1.1504 
1.1918 
1-2349 

1544 
1960 

2393 
2846 
3319 
3814 

1585 

2OO2 

2437 

1626 
2045 
2482 

1667 
2088 
2527 

1708 
2131 

2572 

I75C 
2174 
2617 

792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 
2753 

7    '4    21 
7    »4    « 

8    15    23 

28  3-1 
29  36 
30  38 

1.2799 
1.3270 
1.3764 

2892 
3367 

3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
35H 
4019 

3079 
3564 
4071 

3127 
36i3 
4124 

3175 
3663 
4176 

3222 

3713 
4229 

8    16    23 
8    16    25 
9    '7    26 

3'  39 
33  4' 
34  43 

1.4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9    18    27 

36  45 

56 
57 
58 

1.4826 

1-5399 
1.6003 

4882 
545B 
6066 

4938 
5517 
6128 

4994 
5577 
6191 

5051 
5637 
6255 

5108 

5697 
6319 

5166 
5757 
6383 

5224 
5818 
6447 

5282 
5880 
6512 

5340 
594i 
6577 

o    19   29 

10     20     30 
II      21      32 

38  48 
40  50 
43  53 

59 
60 
61 
62 
63 
64 

1.6643 
1.7321 
1.8040 

6709 
7391 

8115 

6775 
7461 

8190 

6842 
7532 
8265 

6909 
7603 
8341 

6977 
7675 
8418 

7045 
7747 
8495 

7"3 

7820 
8572 

7182 

7893 
8650 

7251 
7966 
8728 

ii    23    34 

12     24     36 
13     26     38 

45  56 
48  60 
51  64 

1.8807 
1.9626 
2.0503 

8887 
9711 
0594 

8967 

9797 
0686 

9047 

9883 
0778 

9128 
9970 
0872 

9210 
6057 
0965 

5292 
0145 
1060 

9375 
6233 

"55 

9458 
0323 
1251 

9542 
0413 

1348 

I4     27     41 

»S    29   44 

16    31    47 

55  68 
58  73 
63  78 

65 
66 
67 
68 
69 
70 
71 

2.1445 

1543 

1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

'7    34    5' 

68  85 

2.2460 
2-3559 
2.4751 

2566 
3673 
4876 

2673 
3789 
5002 

2781 
3906 
5129 

2889 
4023 
5257 

2998 
4142 

5386 

3109 
4262 
5517 

3220 
4383 
5649 

3332 
4504 
5782 

3445 
4627 
59i6 

18    37    55 

20     40     60 

22    43    65 

74  92 
79  99 
87  108 

2.6051 

2.7475 
2.9042 

6187 
7625 
9208 

6325 
7776 
9375 

6464 
7929 
9544 

6605 
8083 
9714 

6746 
8239 
9887 

6889 
8397 
6061 

7034 
8556 
0237 

7179 
8716 

0415 

7326 
8878 
5595 

24    47    71 
26    52    78 
29    58    87 

95  "8 
104  130 
115  M4 

72 
73 
74 

3-0777 
32709 

3.4874 

0961 
2914 
5105 

1146 
3122 
5339 

1334 
3332 
5576 

1524 
3544 
5816 

1716 

3759 
6059 

1910 
3977 
6305 

2106 
4197 
6554 

2305 
4420 
6806 

2506 
4646 
7062 

32    64   96 
36    72  108 

41     82  122 

129  161 
144  k8o 
162  203 

75 
76 
77 
78 

3-7321 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

981 

46    94  139 

186  232 

4.0108 

4-3315 
4.7046 

0408 
3662 
7453 

0713 
4015 
7867 

IO22 

4374 
8288 

1335 
4737 
8716 

1653 
5107 
9152 

1976 
5483 
9594 

2303 
5864 
0045 

2635 
6252 
050^ 

297 
6646 
0970 

53  107  i6c 
62  124  186 
73  146  219 

214  267 
248  310 
292  365 

79 
80 
81 

5.1446 
5-67I3 
6.313^ 

1929 
7297 
3859 

2422 
7894 
4596 

292^ 
8502 
5350 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
640 
7920 

5026 
1066 

8548 

5578 
1742 
9395 

6140 

o1? 

87  175  262 

35<>  437 

026 

82 
83 
84 

7.H54 
8.144. 
9-5I44 

2066 
2636 
9-677 

3002 
3863 
9-845 

3962 
5126 

10.0 

4947 
6427 

10.20 

5958 
7769 
10.39 

6996 

9*5 
10.5 

8062 

0579 
10.78 

9158 
2052 
10.99 

028 

357 

11.20 

Difference  -  col- 
umns cease  to  be 
useful,   owing    to 
the    rapidity  with 
which    the    value 
of  the  tangent 
changes. 

85 
86 
87 
88 

n-43 

11.66 

11.9 

I2.I 

1243 

12.7 

13-00 

13-3° 

13-62 

13-9 

14.30 
19.08 
28.64 

14.67 
19.74 
30.14 

15-06 
20.45 
31-82 

?5-4 
21.2 

33-6 

15.89 

22.02 
35-80 

16.35 
22.90 
38.19 

16.8 
23-8 
40.9 

17-34 
24.90 
44-o 

17-89 
26.03 
47-74 

I8.4 
27.2 
52.0 

89 

57-29 

63.66 

71.62 

81.8 

95-49 

114.6 

143- 

I9I.C 

286. 

573- 

74 


ANTI-LOGARITHMS 


Mats. 

01234 

PROPORTIONAL  PARTS. 

123 

456 

789 

.00 
.0 
.0 
.03 
.04 

1000  1002  1005  1007  1009 
1023  1026  1028  1030  1033 
1047  1050  1052  1054  1057 
1072  1074  1076  1079  1081 
1096  1099  1102  1104  1107 

1012  1014  1016  1019  1021 
1035  1038  1040  1042  1045 
1059  1062  1064  1067  1069 
1084  1086  1089  1091  1094 
1109  1112  1114  1117  1119 

0  0 
0  0 
0  0 
0  0 
0  1 

1  1 
1  1 
1  1 
1  1 
112 

222 
222 
222 
222 
232 

.0 
.06 
.07 
.08 
.09 

1122  1126  1127  1130  1132 
1148  1151  1153  1156  1159 
1175  1178  1180  1183  1186 
1202  1205  1208  1211  1213 
1230  1233  1236  1239  1242 

1135  1138  1140  1143  1146 
1161  1164  1167  1169  1172 
1189  1191  1194  1197  1199 
1216  1219  1222  1225  1227 
1245  1247  1250  1253  1256 

0  1 
0  1 
0  1 
0  1 
0  1 

112 
112 
112 
112 
112 

222 
222 
222 
223 
223 

.1 

.11 
.12 

:12 

1259  1262  1265  1268  1271 
1288  1291  1294  1297  1300 
1318  1321  1324  1327  1330 
1349  1352  1355  1358  1361 
1380  1384  1387  1390  1393 

1274  1276  1279  1282  1285 
1303  1306  1309  1312  1315 
1334  1337  1340  1343  1346 
1365  1368  1371  1374  1377 
1396  1400  1403  1406  1409 

0  1 
0  1 
0  1 
0  1 
0  1 

112 
122 
122 
122 
182 

223 
2    3 
2    3 
2    3 
2    3 

.15 
.16 
.17 
.18 
.19 

1418  1416  1419  1422  1426 
1445  1449  1452  1455  1459 
1479  1483  1486  1489  1493 
1514  1517  1521  1524  1528 
1649  1552  1556  1560  1563 

1429  1432  1435  1439  1442 
1462  1466  1469  1472  1476 
1496  1500  1503  1507  1510 
1531  1535  1538  1542  1545 
1567  1570  1574  1578  1581 

0  1 
0  1 
0  1 
0  1 

1  2  2 
122 
1  2  2 
122 

2    3 
2    3 
2    3 
2    3 
3    3 

•20 

.21 
.22 
.23 

1585  1589  1592  1596  1600 
1622  1626  1629  1633  1637 
1660,  1663  1667  1671  1675 
l^  1702  1706  1710  1714 
1738  1742  1746  1750  1754 

1603  1607  1611  1614  1618 
1641  1644  1648  1652  1656 
1679  1683  1687  1690  1694 
1718  1722  1726  1730  1734 
1758  1762  1766  1770  1774 

0  1 
0  1 
0  1 
0  1 
0  1 

122 
222 
222 
222 

8    3 
3    3 
3    3 
3    4 

.25 
.26 

.27 
.28 
.29 

1778  1782  1786  1791  1795 
1820  1824  1828  1832  1837 
1862  1866  1871  1875  1879 
1905  1910  1914  1919  1923 
1950  1954  1959  1963  1968 

1799  1803  1807  1811  1816 
1841  1845  1849  1854  1858 
1884  1888  1892  1897  1901 
1928  1932  1936  1941  1945 
1972  1977  1982  1986  1991 

0  1 
0  1 
0  1 
0  1 
0  1 

222 
223 
223 
223 
223 

3 
3 
3 
3 
3 

.30 
.31 
.32 
.63 
.84 

1995  2000  2004  2009  2014 
2042  2046  2051  2056  2061 
2089  2094  2099  2104  2109 
2138  2143  2148  2153  2158 
2188  2193  2198  2203  2208 

2018  2023  2028  2032  2037 
2065  2070  2075  2080  2084 
2113  2118  2123  2128  2133 
2163  2168  2173  2178  2183 
2213  2218  2223  2228  2234 

0  1 
0  1 
0  1 
0  1 

1  1 

223 
223 
223 

3    4 
3    4 
3    4 

.35 
.86 
.87 
.88 
.39 

2239  2244  2249  2254  2259 
2291  2296  2301  2307  2312 
2344  2350  2355  2360  2366 
2399  2404  2410  2415  2421 
2455  2460  2466  2472  2477 

2265  2270  2275  2280  2286 
2317  2323  2328  2333  2339 
2371  2377  2382  2388  2393 
2427  2432  2438  2443  2449 
2483  2489  2495  2500  2506 

119 

1  1  2 
112 

233 

445 

.40 
.41 
.42 
.43 
.44 

2512  2518  2523  2529  2535 
2570  2576  2582  2588  2594 
2630  2636  2642  2649  2655 
2692  2698  2704  2710  2716 
2754  2761  2767  2773  2780 

2541  2547  2553  2559  2564 
2600  2606  2612  2618  2624 
2661  2667  2673  2679  2685 
2723  2729  2735  2742  2748 
2786  2793  2799  2805  2812 

112 
112 
112 
112 
112 

234 

455 

234 

456 

45 

.46 
.47 
.48 
.49 

2818  2825  2831  2838  2844 
2884  2891  2897  2904  2911 
2951  2958  2965  2972  2979 
3020  3027  3034  3041  3048 
090  3097  3105  3112  3119 

2851  2858  2864  2871  2877 
2917  2924  2931  2938  2944 
2985  2992  2999  3006  3013 
3055  3062  3069  3076  3083 
3126  3133  3141  3148  3155 

112 
1  1  2 
112 
112 
112 

834 
334 
384 
344 
344 

556 
556 
566 
666 
666 

ANTI-LOGARITHMS 


75 


Hants. 

01234 

56789 

PROW 

>RTIONAL  F 

ARTS. 

J.  2  3 

456 

789 

.50 

.51 
52 

3162  3170  3177  3184  3192 
3236  3243  3251  3258  3266 
3311  3319  3327  3334  3342 

3199  3206  3214  3221  3228 
3273  3281  3289  3296  3304 
3350  3357  3365  3373  3381 

112 
122 
122 

.53 
.54 

3388  3396  3404  3412  3420 
3467  3475  3483  3491  3499 

3428  3436  3443  3451  3459 
3508  3516  3524  3532  3540 

122 

.55 

3548  3556  3565  3573  3581 

3589  3597  3606  3614  3622 

.56 
.57 
.58 

3631  3639  3648  3656  3664 
3715  3724  3733  3741  3750 
3802  3811  3819  3828  3837 

3673  3681  3690  3698  3707 
3758  3767  3776  3784  3793 
3846  3855  3864  3873  3882 

123 
1  2  3 

445 

678 

.60 

.61 

3981  3990  3999  4009  4018 
4074  4083  4093  4102  4111 

4027  4036  4046  4055  4064 
4121  4130  4140  4150  4159 

1  2 

.62 
.63 
.64 

4266  4276  4285  4295  4305 
4365  4375  4385  4395  4406 

4315  4325  4335  4345  4355 
4416  4426  4436  4446  4457 

1  2 
1  2 

.66 
.67 

4571  4581  4592  4603  4613 
4677  4688  4699  4710  4721 

4624  4634  4645  4656  4667 
4732  4742  4753  4764  4775 

1  2 
1  2 

456 

7  9  10 

.68 
.69 

4786  4797  4808  4819  4831 
4898  4909  4920  4932  4943 

4842  4853  4864  4875  4887 
4955  4966  4977  4989  5000 

1  2 
1  2 

567 

8  9  10 

.71 

5129  5140  5152  5164  5176 

5188  5200  5212  5224  5236 

1  2 

567 

8  10  11 

.73 

5370  6383  5395  5408  6420 

5433  5445  5458  5470  5483 

1  3 

568 

9  10  11 

.75 

.76 

5623  5636  5649  5662  5675 
5754  5768  5781  5794  5808 

5689  6702  6716  6728  6741 
6821  5834  5848  5861  5876 

9  10  IS 
9  11  12 

.78 
.79 

6026  6039  6053  6067  6081 
6166  6180  6194  6209  6223 

6095  6109  6124  6138  6152 
6237  6252  6266  6281  6295 

10  11  13 
10  11  13 

.80 
.81 
.82 

6310  6324  6339  6353  6368 
6457  6471  6486  6501  6516 
6607  6622  6637  6653  6668 

6383  6397  6412  8427  6442 
6531  6546  6561  6577  6592 
6683  6699  6714  6730  6745 

134 

235 

6  7  9 
689 

10  12  13 
11  12  14 
11  12  14 

.83 
.84 

6761  6776  6792  6808  6823 
6918  6934  6950  6966  6982 

6839  6855  6871  6887  6902 
6998  7015  7031  7047  7063 

235 

6  8  10 

11  13  14 
11  13  15 

.85 

7079  7096  7112  7129  7146 

7161  7178  7194  7211  7228 

236 

7  8  10 

12  13  15 

.87 
.88 
.89 

7413  7430  7447  7464  7482 
7586  7603  7621  7638  7656 
7762  7780  7798  7816  7834 

7499  7516  7534  7551  7568 
7674  *S91  7709  7727  7745 
7852  7870  7889  7907  7926 

236 
245 
245 

7  9  10 
7  9  11 
7  911 

12  14  16 
12  14  16 
18  14  16 

.90 

.91 
.92 

.93 
.94 

7943  7962  7980  7998  8017 
8128  8147  8166  8185  8204 
8318  8337  8356  8375  8395 
8511  8531  8651  8570  8590 
8710  8730  8750  8770  8790 

8222  8241  8260  8279  8299 
8414  8433  8453  8472  8492 
8610  8630  8650  8670  8690 
8810  8831  8851  8872  8892 

246 
246 
246 
246 

8  9  11 
8  10  12 
8  10  12 
8  10  12 

13  15  17 
14  15  17 
14  16  18 
14  16  18 

.95 

.96 
.97 
.98 
.99 

8913  8933  8954  8974  8995 
9120  9141  9162  9183  9204 
9333  9354  9376  9397  9419 
9550  9572  9594  9616  9638 
9772  9795  9817  9840  9863 

9016  9036  9057  9078  9099 
9226  9247  9268  9290  9311 
9441  9462  9484  9506  9528 
9681  9683  9705  9727  9750 
9886  9908  9931  9954  9977 

246 
246 
247 
247 
267 

8  10  12 
8  11  13 
9  11  13 
9  11  13 
9  11  14 

15  17  19 
15  17  19 
15  17  20 
16*18  20 
16  1820 

76 


LOGARITHMS. 


10 

0 

1 

2 

3 

4 

•5 

6 

7 

8 

9 

123 

456 

789 

oooo 

0043 

0086 

OI2S 

0170 

0212 

0253 

0294 

0334 

0374 

Use  Table  on  p.  58 

11 

12 
13 
14 
15 
16 
17 
18 
19 

0414 
0792 

"39 

0453 
0828 

"73 

0492 
0864 
1206 

0531 

0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1  004 
1335 

0682 
1038 
1367 

6719 
1072 
1399 

0755 
1106 
T43Q 

4  8  ii 
3  7  10 
3  6  10 

15  19  23 

14    17    21 

13  16  19 

26  30  34 
24  28  31 
23  26  29 

1461 
1761 
2041 

1492 
1790 
2068 

1523 
1818 
2095 

1553 
1847 

2122 

1584 

1875 
2148 

1614 
1903 

2175 

1644 
!93i 

2201 

1673 

1959 
2227 

i7<>3 
1967 

2253 

1732 

20  1.  } 

2279 

369 
368 
5  8 

12  15  18 

II  14  17 
ii  13  16 

21    24    27 

20   22    25 

18  21  24 

2304 

2553 
2788 

2330 
2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
29OO 

2455 
2695 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 

2765 
2989 

5  7 
5  7 
4  7 

10    12    15 

9  12  14 
9  «  i3 

17    2O   22 

16  19  21 

16  18  20 

20 
21 
22 
23 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

4  6 

8  ii  13 

15  17  *9 

3222 

3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3^74 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
356o 
3747 

3385 
3579 
3766 

3404 

3598 
3784 

4  6 
4  6 

4  6 

8    10    12 
8    IO    12 

79" 

14  16  18 
14  15  1.7 
13  "5  *7 

24 
26 
26 

3802 
3979 
4!*> 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 

4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 

4133 
4298 

4   5 
3   5 
3   5 

7    9  ii 

7    9  1° 
7    8  .10 

12  14  16 

12   14   15 
II    13    15 

27 
28 

29 

43M 
4472 
4624 

4330 
1487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 

4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 

4609 
4757 

3   5 
3   5 
3  4 

689 
689 
679 

II    13    14 
II    12    14 
10    12    13 

30 
31 
32 
33 
34 
35 
36 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3   4 

679 

10  ii  13 

4914 
5051 

5185 

4928 
5065 
5198 

4942 

5079 
5211 

4955 
5092 
5224 

4969 
5105 

5237 

4983 
5"9 
5250 

4997 
5132 
5263 

5011 

5145 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

3  4 
3  4 
3  4 

678 
5    7    8 
5    6    8 

10    II    12 

9  ii  12 
9  10  12 

5315 
5441 
5563 

5328 
5453 
5575 

5340 
5465 

5587 

5353 
5478 
5599 
5717 
5832 
5944 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

54i6 

5539 
5658 

5428 

5551 
5670 

3  4 
4 

4 

5    6    8 
5    6    7 
5    6    7 

9  10  ii 
9  jo  u 
8  10  ii 

37 
38 
39 

5682 
5798 
59TI 

5694 
S&og 
5922 

5705 
5821 
5933 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 

5877 
5988 

5775 
5888 

5999 

5786 

5899 
6010 

3 
3 
3 

5    6    7 
5    6    7 
457 

8    9  10 
8    9  jo 

8    9  10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

3 

4     5    6 

8    9  10 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 
6345 
6444 
6542 
6637 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

3 
3 
3 

456 
4    5    6 
4     5    6 

7    »    9 
7    8    9 
7    8    q 

44 
45 
46 
"47 
48 
49 

6435 
6532 
6628 

6454 
655i 
6646 

6464 
6561 
6656 

6474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 

6599 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

3 
3 
3 

5    6 
5    6 
5    6 

7    8    9 
7     8    g 

7     7    8 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

3 
3 
3 

5     5 
4     5 
445 

6    7    8 
6    7    fe 
6     7    8 

50 

51 
52 
53 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

3 

345 

6     7     8 

7076 
7160 
7243 

7084 
7168 
7251 

7093 
7177 

7259 

7101 

7185 
7267 

7110 
7193 

7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7M3 
7226 
7308 

7152 
7235 
7316 

3 

2 
2 

345 
345 
345 

6    7     8 
6    7     7 
667 

667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

122 

345 

LOGARITHMS 


77 


55 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

7404 

7412 

7.419 

7427 

7435 

7443 

745i 

7459 

7466 

7474 

122 

3    4     5 

5  6  7 

56 
57 
58 

7482 
7559 
7634 

7490 
7566 
7642 

7497 
7574 
7649 

7505 

7582 
7657 

7513 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

7551 
7627 
7701 

122 
I  2  2 
I  I  2 

345 
345 

344 

5  6  7 
5  6  7 
5  6  7 

59 
60 
61 

7709 

7782 
7853 

77i6 
7789 
7860 

7723 
7796 
7868 

773i 
7803 
7871 

7738, 
7810 
7882 

7745 
7818 
7889 

7752 
7825 
7896 

7760 
7832 
7903 

7767 

7839 
7910 

7774 
7846 
79^7 

I  I  2 
112 
112 

344 
344 
344 

5  6  7 

5  6  C 
5  6  6 

62 
63 
64 

7924 

7993 
8062 

793i 
Sooo 
8069 

7938 
8007 
8075 

7945 
8014 
8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 

8048 
8116 

7987 
805^ 
8122 

I  1  2 
112 

I  I  2 

112 

334 
334 
334 

5  6  6 

5  S  6 

s  s  e 

65 
66 
67 
68 

8129 

8i95 
8261 

8325 

8136 

8202 
8267 
8331 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

3     3     4    5     5    C 

8209 

8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

112 
112 
I  I  2 

334 
334 
334 

5  5  6 
5  5  6 
4  5  6 

69 
70 
71 

8388 
8451 
8513 

8395 
8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 

8555 

8439 
8500 
8561 

8445 
8506 
8567 

1  I  2 
I  I  2 
112 

234 
234 

234 

4  5  6 
4  5  (' 
455 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

859J 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 

8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

112 
J  I  2 
112 

234 

2     3     4 
234 

455 
455 
455 

75 

875i 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

I  I  2 

233 

4  5  5 

76 
77 
78 
79 
80 
81 

8808 
8865 
8921 

8814 
8871 
8927 
8982 
9036 
9090 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 
9004 
9058 
9112 

8842 
8899 
8954 

8848 
8904 
8960 

9015 
9069 
9122 

8854 
8910 
8965 

S85S) 
8915 
897] 

I  I  2 

233 

455 

I  I  2 

233 

445 

8976 
9031 
9085 

8987 
9042 
9096 

8993 
9047 
9101 

8998 

9°53 
9106 

9009 
9063 
9117 

9020 
9074 
9128 

9025 
9079 
9133 

1  I  2 
1  I  2 

233 
233 

445 

445 

82 
83 
84 

9138 
9191 
9243 

9T43 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9r59 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 
9279 

9180 
9232 
9284 

9186 
9238 
9289 

112 

2    3     3:  4     4     5 

112 

233 

445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

233 

86 
87 
88 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

936o 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

938o 
9430 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

0  I  I 

223 

3  4  -1 

89 
90 
91 

9494 
9542 
959° 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

95i8 
9566 
9614 

9523 
957' 
9619 

9528 
9576 
9624 

9533 
958i 
9628 

9538 
9580 
9633 

Oil 
Oil 

223 
223 

344 
344 

92 
93 
94 

9638 
9685 
9731 

9f>43 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 
979' 

9657 
9703 
9750 

9661 
9708 
9754 

9666 
97*3 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

Oil 
Oil 
0  I  I 

223 
223 
2-23 

344 
344 
3  4  4 

95 

9777 

9782 

9786 

9795 

9800 

9805 

9809 

9814 

9818 

Oil 

2       2       3 

344 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 

9934 

9850 
9894 
9939 

9854 
9899 

9943 

9859 
9903 
9948 

9863 
9908 
9952 

Oil 

0  I  I 
Oil 

223 

223 
2       2       3 

344 

344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

999  i 

9996 

O  I  1 

223 

334 

78 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

ooooo 

043 

087 

130 

173 

217 

260 

303 

346 

389 

101 
102 
103 

432 
860 

oi  284 

•  475 
903 
326 

5i8 
945 
368 

56i 
988 
410 

604 
030 
452 

647 
072 
494 

689 
H5 
536 

732 
157 

578 

775 
199 
620 

817 
242 

662 

104 
105 
106 

703 

O2  IIQ 

531 

745 
160 
572 

787 
202 
612 

828 
243 
653 

870 
284 
694 

912 
325 

735 

953 
366 
776 

995 
407 
816 

036 

449 

857 

078 
490 
898 

107 
108 
109 

938 
03342 

743 

979 

383 
782 

019 

423 
822 

060 

463 
862 

IOO 

503 
902 

141 

543 
941 

181 

583 
981 

222 
623 
O2  1 

262 
663 
060 

302 
703 

IOO 

To  find  the  logarithm  of  a  number:  First,  locate  in  the 
table  the  mantissa  which  lies  in  line  with  the  first  two  figures  of  the 
number  and  underneath  the  third  figure,  then  increase  this  mantissa 
by  an  amount  depending  upon  the  fourth  figure  of  the  number  and 
found  by  means  of  the  interpolation  columns  at  the  right;  secondly, 
determine  the  characteristic,  or  the  exponent  of  that  integer  power 
of  10  which  lies  next  in  value  below  the  number;  for  example, 
log  600- 0.7782  f  2.;  log  73.46=0.8661  +  1.; 

log  .006=0.7782-3.;  log  .7346=0.8661-1.; 

log  6.003=  0.7784  +  0. ;  log  7349=  0.8662  +  3. 

The  logarithm  of  a  product  of  two  or  more  numbers  is  the  sum  of 
the  logarithms  of  its  factors;  for  example, 

log.  (.0821  X  463.2)=  (0.9143  -2.)  +  (0.6658  +  2.)  =  0.5801  +  1. 
The  logarithm  of  a  quotient  is  the  difference  between  the  logar- 
ithms of  the  dividend  and  divisor;  for  example, 

log.  (.5321 -916)=  (0.7260-1.)  -  (0.9619 +  2.)  =  0.7641-4. 
The  logarithm  of  a  power  or  root  of  a  number  is  the  exponent 
times  the  logarithm  of  the  number;  for  example, 

log  V.863)3=3/2  X  (0.9360— 1.)  =  0.9040 -1. 

To  find  the  number  from  its  logarithm:  Locate  in  the  table 
the  mantissa  next  less  than  the  given  mantissa,  then  join  the  figure 
standing  above  it  at  the  top  of  the  table  to  the  two  figures  at  the 
extreme  left  on  the  same  line  as  the  mantissa,  and  finally  to  these 
three  join  the  figure  at  the  top  of  the  interpolation  column  which 
contains  the  difference  between  the  two  mantissae.  In  the  four- 
figure  number  thus  found,  so  place  the  decimal  point  that  the 
number  shall  be  the  product  of  some  number,  that  lies  between 
1  and  10,  by  a  power  of  10  whose  exponent  is  the  characteristic 
of  the  logarithm.  For  example, 

antilog  (0.6440 +  3)  =  4405; 
antilog  (0.3069 -2)  =  .02027. 

Caution.  In  adding  and  subtracting  logarithms  it  is  well  to 
remember  that  the  mantissa  is  always  essentially  positive  and  may 
or  may  not  therefore,  have  the  same  sign  as  its  characteristic. 


INDEX 


Absolute,  expansion  of 

liquid   -         -         -         -  36 

Accuracy  of  measurement  8 

Air,  buoyancy,  correction 

for  in  weighing      -         -  15 

Air,  density  of                    -  16 

Alcohol,  fuel  value  of        -  53 

Apparent  expansion           -  36 

of  alcohol      -                  -  33 

Archimedes'  principle       -  15 

Balance,  sensitive     -         -  12 
Boyle's  Law,  Applied  16, 19,  21 

Calorimetry       -  44 

Calorimeter      -         -         -  44 

Capillarity,  rise  of  liquids 

in  tubes        -        -         -  25 

Carbon    dioxide,    relative 

density  of     -                  -  19 

Charles'  Law  -        -  *     -  18 

Coefficient  of  Expansion 
of  liquid  by  Regnault's 
method                            -  36 
by  pycnometer  method  -  38 
of  glass  by  weight-ther- 
mometer method         -  42 

Contact,  angle  of      -         -  25 
Cooling  curve            -         -49 

Cooling,  law  of                   -  48 

Data  sheets  7 


Density,   of  air,  determi- 
nation of,                         -  16 
of  carbon  dioxide,  deter- 
mination of            -         -  19 
of  a  cylindrical  solid     -  12 

Double  Weighing     -         -  15 

Errors      -                            -  7 

Expansion, 

absolute,  of  liquid          -  36 

Experiments,  list  of           -  6 

Figures,  significant             -  8 

Force  Table  58 

Fuel  value  of  alcohol         -  53 

Glass,  coefficient  of  expan- 
sion     -         -         -         -  42 


Heat  of  Fusion,  Wood's 

Alloy    - 
Hooke's  Law    - 


-  50 

-  31 

-  54 


Junker's  Calorimeter 

Logarithms,  table  of 

4  place        -  -  76 

Measurements,     precision 

of  -  8 

Method  of  cooling  -  48 

of  Components      -  .    -  58 

of  heating     -  -  46 

of  mixtures  -        -  -  50 

of  vibrations    -  -     -  12 

pycnometer           -  -  38 

weight-thermometer      -  42 

Parallelogram  Law  -  58 

Percent  error             -  -  10 


80 


INDEX 


Plotting  of  Curves   -          -  41 

Physical  Tables  60 

Principle  of  Moments       -  55 

Pycnometer,  expansion  of 

liquid  by                           -  38 
Radiation,  correction  for 

rate  of                              -  52 

Reference  books        -         -  6 

Rest  point,  of  balance       -  12 

Rider,  use  of  the      -         -  13 

Sensitiveness    of    balance 

defined          -   ^     -         -  13 

Significant  Figures  -  8 

Specific   heat   of   a   liquid 

by    method    of   heating  46 

by    method    of    cooling  48 

Surface  tension  by  Jolly's 

balance                            -  23 

in   capillary  tubes         -  25 


Tension,    surface,  by    di- 
rect measurement          -     23 

Thermometer,  comparison 
of  alcohol  and  water      -     33 

Vibration,  method  of  12 

Viscosity,   coefficient  of  -     28 
Volumenometer,  the          -     20 

Water,    equivalent    of    a 
body     -  -     45 

equivalent    of    a    ther- 
mometer      -  -     45 
equivalent  of  a  glass 
calorimeter  -  45 

Weighing,    method    of 

double  -     15 

by  method  of  vibrations     12 


Weight  thermometer 


42 


Young's  Modulus,  by 

stretching     -  -     31 


PHYSICAL   MEASUREMENTS 


MINOR 


PART  II.    HEAT,  MECHANICS  AND 

PROPERTIES  OF  MATTER 

1917 


PHYSICAL   MEASUREMENTS 

A  LABORATORY  MANUAL  IN  GENERAL  PHYSICS 
FOR  COLLEGES 

In  Four  Parts 
BY 

RALPH  S.  MINOR,  Ph.  D. 

Associate  Professor  of  Physics,  University  of  California 

PART  II 
HEAT,  MECHANICS  and  PROPERTIES  of  MATTER 

In  collaboration  with 
Wendell  P.  Roop,  A.  B. 

and 
Lloyd  T.  Jones,  Ph,  D. 

Instructors  in  Physics,  University  of  California 

BERKELEY,  CALIFORNIA 
1917 


Copyrighted  in  the  year  1917  by 
Ralph  S.  Minor 


Wetzel  Bros.  Printing  Co. 

2110  Addison  Street 
Berkeley,  Calif. 


LIST  OF  EXPERIMENTS 

Page 

26.  Uniformly  Accelerated  Motion  -  5 

27.  Centripetal  Force      -  8 

28.  The  Simple  Pendulum  -  10 

29.  Momentum       -  -  14 

30.  Moment  of  Inertia    -  -  15 

31.  Mechanical  Equivalent  of  Heat  by  Calendar's 
Method    -  -  18 

32.  Efflux  of  Gases  -  31 

33.  Calibration  of  a  set  of  weights  -  23 

34.  Absolute  Calibration  of  a  Thermometer  -         -  25 

35.  Heat  of  Solution                           -         -  -         -  31 

36.  Heat  of  Neutralization  -  33 

37.  Variation  of  Boiling  Point  with  Pressure  -         -  35 

38.  Constant  Volume  Air-Thermometer  -  37 

39.  Vapor  Pressure  and  Temperature       -  -  39 

40.  Hygrometry      -  -  41 

41.  Ratio  of  the  Two  Specific  Heats  of  Air  -          -  44 

42.  The  Heat  of  Fusion  of  Tin         -         -  -  46 


REFERENCE  BOOKS 

Duff:  Text-Book  of  Physics  (Fourth  Edition). 

Edser:  Heat  for  Advanced  Students. 

Ganot:  Text-Book  of  Physics  (18th  Edition). 

Hastings  and  Beach:  General  Physics. 

Kimball:  College  Physics. 

Preston:  Theory  of  Heat  (Second  Edition). 

Reed  and  Guthe:  College  Physics. 

Spinney:  Text-Book  of  Physics. 

Watson:  Text-Book  of  Physics  (Fourth  Edition,  1903). 

Kaye  and  Laby:  Physical  and  Chemical  Constants. 
Landolt  and  Bornstein:  Physical  and  Chemical  Tables. 
Smithsonian  Institute:  Physical  Tables. 


HEAT, 
MECHANICS   and   PROPERTIES   of   MATTER 

26.     UNIFORMLY  ACCELERATED  MOTION. 

To  determine  the  ratio  of  the  weight  unit  of  force  to  the 

absolute  unit  of  force. 

When  the  body  is  acted  upon  by«  a  constant  force, 
its  motion  is  uniformly  accelerated.  The  force  required 
to  produce  a  given  acceleration  is  directly  proportional 
to  the  mass  on  which  it  acts  and  to  the  acceleration 
produced  in  the  mass. 

When  a  body  moves  under  the  influence  of  its  own 
weight,  it  moves  with  a  certain  characteristic  accelera- 
tion, which  is  the  same  for  all  bodies  in  any  given  locality. 
If  large  bodies  are  acted  on  by  proportionally  large 
forces,  their  large  mass  operates  to  prevent  the  increased 
acceleration  which  would  otherwise  occur.  Weight- 
force  is  rigorously  proportional  to  mass. 

When  a  body  moves  under  the  influence  of  some  force 
other  than  weight,  the  acceleration  produced  is  propor- 
tional to  the  force  acting.  In  weight  units,  the  force 
exerted  on  unit  mass  by  the  earth  is  called  unit  force. 
In  unit  mass,  it  produces  an  acceleration  equal  to  the 
characteristic  acceleration  produced  by  weight.  In 
absolute  units,  the  force  producing  unit  acceleration  in 
unit  mass  is  called  unit  force,  (the  dyne).  The  ratio 
of  the  weight  unit  to  the  absolute  unit  is  numerically 
equal  to  the  acceleration  produced  by  weight-force. 

In  this  experiment,  the  acceleration  produced  in  a 
given  body  by  a  given  force  will  be  directly  measured, 
and  the  force  producing  the  acceleration  calculated  in 


6  UNIFORMLY  ACCELERATED  MOTION  [26 

absolute  units.     Knowing  the  accelerating  force  in  weight 
units,  the  ratio  of  the  units  may  be  calculated. 

The  acceleration  is  to  be  measured  by  a  device  by 
means  of  which  the  body,  as  it  moves,  leaves  a  trace 
which  enables  us  to  infer  the  position  of  the  body  at  any 
instant  during  its  motion.  Two  forms  of  apparatus  are 
used.  In  one,  the  moving  body  is  a  tuning-fork  which 
traces  a  wavy  line  on  a  fixed  whitened  glass  plate.  In 
the  other,  the  fork  is  stationary  and  the  plate  moves. 
In  both,  the  body  moves  vertically,  between  guides. 
The  body  may  fall  freely,  under  the  influence  of  its 
weight  alone;  or,  by  means  of  a  pulley  and  counter- 
weight, the  force  acting  may  be  reduced  to  any  desired 
value. 

(a)  Paint  the  plate  with  a  thin  coat  of  corn-starch, 
mixed  with  alcohol.  The  alcohol  will  evaporate  and 
leave  a  surface  on  which  the  trace  may  be  taken.  Level 
the  apparatus  carefully.  If  the  body  does  not  move 
vertically,  friction  in  the  guides  will  cause  error. 

Adjust  the  stylus  on  the  fork  so  that  it  presses  very 
lightly  against  the  plate.  If  the  stylus  does  not  press 
tightly  enough,  parts  of  the  trace  may  be  missing.  But 
if  it  presses  too  hard,  friction  will  interfere  with  the 
proper  motion  of  the  stylus. 

The  traces  to  be  taken  are  five  in  number.  If  con- 
ditions are  favorable,  all  may  be  obtained  on  one  plate, 
and  in  case  this  is  possible,  postpone  the  measurement 
of  the  traces  until  all  of  them  are  ready.  Otherwise, 
the  measurements  must  be  taken  before  the  plate  is 
prepared  for  new  traces. 

First  obtain  a  trace  with  the  body  falling  freely.  To 
be  usable,  the  trace  must  be  distinct  over  a  distance  of 
at  least  30  to  40  cm. 


26]  UNIFORMLY  ACCELERATED  MOTION  7 

For  the  second  trace  use  a  counterweight  weighing 
300  grams  less  than  the  falling  fork  or  plate;  then,  in 
succession,  others  giving  accelerating  forces  of  200  and 
100  grams. 

Finally,  by  using  all  the  counterweights,  reduce  the 
acceperating  weight-force  to  zero.  Give  the  moving 
system  a  push,  and  take  a  fifth  trace  showing  the  negative 
acceleration  due  to  friction.  If  the  friction  were  not 
acting,  all  the  observed  accelerations  would  be  greater. 

(b)  In  measuring  the  traces,  proceed  as  follows. 
Mark  the  crests  of  a  convenient  odd  number  of  consecutive 
waves  near  the  beginning  and  also  near  the  end  of  the 
trace.  Lay  the  meter  stick  on  the  trace  and  read  off  the 
positions  of  the  crests  which  were  marked.  Count  the 
number  of  unmarked  crests  between  those  whose  posi- 
tions are  noted.  Record  the  vibration  frequency  marked 
on  the  fork. 


(c)  Obtain  the  mean  velocity  for  the  intervals  at  the 
beginning  and  end  of  each  trace  by  dividing  the  length 
of  the  interval  by  the  corresponding  time.     This  time 
interval  is  calculated  from  the  number  of  wave-lengths 
in  the  interval  and  the  frequency  of  the  fork. 

Determine  the  number  of  wave-lengths  lying  between 
the  centers  of  the  two  intervals.  The  change  of  velocity 
during  the  time  interval  between  these  two  middle 
points,  divided  by  the  time  interval,  gives  the  change 
in  velocity  per  second,  or  the  acceleration. 

(d)  Make    out    a    table,     containing    the    following 
columns:    (1)  acceleration;  (2)  acceleration,  corrected  for 


8  CENTRIPETAL  FORCE  [26-27 

effect  of  friction;  (3)  mass  of  body  accelerated;  (4)  force 
acting  on  body,  in  absolute  units;  (5)  force  acting  on 
body,  in  weight  units 

(e)  By  comparing  your  different  values  for  the  ratio 
of  the  two  units  with  their  mean,  estimate  the  probable 
error  of  your  work. 

27.     CENTRIPETAL  FORGE. 

References. — Duff,  pp.  23,  24,  25;  Kimball,  pp.  75,  76. 

To  determine  the  force  necessary  to  keep  a  body  of  given 
mass  in  a  circle  of  given  radius,  while  it  moves  with  con- 
stant speed. 

Experience  shows  that  a  body  in  motion  will  continue 
to  move  with  the  same  speed  in  the  same  straight  line, 
unless  acted  upon  by  some  outside  force.  An  outside 
force,  if  acting  in  the  direction  of  the  motion,  will  cause 
a  change  in  speed;  if  acting  at  right  angles  to  the  direction 
of  motion,  it  will  cause  no  change  in  speed,  but  will 
cause  a  change  in  the  direction  of  the  motion.  A  body 
in  motion  always  moves  in  a  straight  line,  unless  there 
is  a  force  applied  causing  it  to  leave  the  straight  line.  If 
the  force  perpendicular  to  the  line  of  the  motion  be  momen- 
tarily supplied,  the  direction  of  the  motion  is  changed, 
but  the  body  afterward  moves  in  a  straight  line  at  an 
angle  with  its  former  direction.  If  the  force  be  continu- 
ously applied,  the  body  moves  in  a  curved  path.  If 
the  body  be  kept  in  a  circular  path,  a  force  of  definite 
magnitude  must  be  continuously  applied  to  the  body, 
the  direction  of  the  force  being  always  perpendicular  to 
the  instantaneous  direction  of  the  motion.  Since  the 
instantaneous  direction  of  motion  is  along  the  tangent, 
the  force  perpendicular  to  the  direction  of  motion  must  be 


27]  CENTRIPETAL  FORCE  9 

along  the  radius  of  the  circle.  If  the  force  ceases  to  be 
applied,  the  body  ceases  to  leave  the  straight  line  and 
hence  continues  to  move  in  the  tangent  to  the  circle  at 
the  position  occupied  by  the  body  at  the  instant  the 
force  ceased  to  act. 

The  force,  /,  required  to  maintain  motion  in  a  circle 
of  radius,  r,  of  a  mass,  w,  moving  with  rim  speed,  v,  and 
angular  speed,  co,  is 

f  =  k  m  v2  /  r,      or  since  co  r  =  v,      f  =  k  m  co2  r. 

If  the  quantities  m,  v,  and  r  are  expressed  in  C.  G.  S.  units 
the  factor  k  will  be  unity  and  the  force  will  be  given  in 
dynes. 

To  a  rotator  is  attached  the  "centripetal  force"  appar- 
atus. Two  masses,  nti  and  w2,  are  arranged  to  slide  along 
the  horizontal  guides.  One  of  these,  m\,  is  attached,  by 
means  of  cords  passing  over  pulleys,  to  a  large  mass  M, 
which  can  slide  up  and  down  along  the  vertical  rod.  As 
the  speed  of  rotation  is  increased,  more  and  more  force 
must  be  supplied  in  order  to  hold  it  to  a  circular  path. 
Finally,  when  the  speed  passes  a  certain  value,  the 
force  necessary  to  keep  the  mass  moving  in  its  circu- 
lar path  is  greater  than  the  weight  of  M  can  supply, 
so  the  mass  M  is  lifted.  Its  weight,  Mg  dynes,  represents 
the  normal  or  centripetal  force  it  supplies  to  the  mass  m\. 

(a)  To  determine  the  radius  for  m\,  measure  the  dis- 
tance from  the  center  of  rotation  to  the  center  of  the 
mass  wi  when  the  mass  M  is  against  the  lower  stop  and 
again  when  it  is  against  the  upper  stop  and  take  the 
average  of  these  distances  as  the  radius  r\.  Fix  m2  at  a 
distance  r\  from  the  center  of  rotation.  Note  the  value 
of  the  masses. 

The  speed  is  to  be  calculated  from  a  determination  of 
the  number  of  revolutions  in  a  measured  time.  This 


10  THE  PRINCIPLE  OF  MOMENTS  [27-28 

may  be  most  accurately  determined  by  measuring  the 
time  required  for  some  even  number  of  hundreds  of  revo- 
lutions, starting  and  stopping  the  stop-clock  while  the 
apparatus  is  in  motion.  To  eliminate  friction  the  speed 
should  be  so  regulated  that  the  mass  M  slowly  rises  and 
then  falls  between  the  stops  continuously  during  a  trial, 
which  should  occupy  from  three  to  five  minutes. 

Make  two  more  determinations  of  the  speed  with  the 
masses  at  the  same  distance. 

(b)  Make  several  additional  trials,  selecting  each  time 
a  different  value  of  the  mass  M,  or  of  the  distance  of  mi 
and  w2  from  the  axis. 


(c)  For  each  trial,  test  the  equality  of  the   weight, 
(Mg),  and. the  calculated  centripetal  force  required,  and 
determine   the   percentage    difference   from    the    weight. 

(d)  Point  out  the  principal   sources  of  error  in  the 
experiment. 

28.     THE  SIMPLE  PENDULUM. 

Reference. — Duff,  pp.  78,  82. 

To  determine  the  acceleration  due  to  gravity  from  the 
observed  period  and  length  of  a  simple  pendulum. 

If  a  conical  pendulum  and  a  simple  pendulum  have  the 
same  length,  /,  and  each  is  oscillating  through  a  small 
amplitude  (2°  or  less)  the  periods  are  equal.  The  projec- 
tion of  the  motion  of  the  conical  pendulum  on  any 
diameter  corresponds  to  that  of  the  simple  pendulum. 
If  r  is  the  radius  of  the  circle  the  conical  pendulum 
describes,  the  force  toward  the  center  is  constant  and 


28] 


THE  SIMPLE  PENDULUM 


11 


Fig.  1.  —  Showing  a  conical  pendulum  (a)  and  a  simple  pendulum  (b) 
of  the  same  length. 

its  value  is  m  v*  /  r  or  m  co2  r,  where  v  is  its  linear  and  co 
its  angular  velocity.  In  t  seconds  after  the  conical 
pendulum  has  passed  one  end  of  the  diameter  shown  in 
(a)  the  angle  turned  through  is  co/  and  the  component 
of  the  centripetal  force  along  this  diameter  will  be 
m  co2  r  cos  co  t.  Let  x  be  the  projection  of  r  on  the 
diameter  at  this  instant.  Then  cos  co  t  =  x  /  r,  and  the 
component  of  the  force  along  the  diameter  becomes 
m  co2  x. 

When  the  simple  pendulum  is  an  equal  distance,  x, 
from  its  rest  position  the  force  urging  it  toward  its  rest 
position  is  /  =  m  g  x  /  I.  If  the  two  pendulums  have 
the  same  mass,  m,  these  forces  must  be  equal.  Then 


m 


g  x  /  /,  or  co2  =  g  /  I. 


In  any  circular  motion  of  period  7,  coT  =  2w. 
Substituting  this  value  of  co  in  the  preceding  equation 
and  solving  for  T  we  have 

T  -  27T 


12  THE  SIMPLE  PENDULUM  128 

Method  of  Coincidences. 

In  the  present  experiment,  T  is  to  be  measured  by 
comparing,  by  the  "method  of  coincidences,"  the  period 
of  the  simple  pendulum  with  that  of  a  clock  pendulum  of 
known  period.  An  electric  circuit  is  completed  through 
an  electric  bell,  the  clock  pendulum,  the  simple  pendulum, 
and  the  contacts  at  the  bottom  of  each  pendulum.  As- 
sume that  the  period  of  the  clock  pendulum  is  two  seconds, 
that  is,  that  the  time  of  a  single  swing  or  half-vibration 
is  one  second.  If  the  period  of  the  simple  pendulum  were 
the  same  and  the  two  pendulums  be  started  together, 
they  would  vibrate  in  coincidence  and  the  bell  would 
ring  with  every  passage.  If,  however,  the  time  of  a  single 
swing  of  the  simple  pendulum  were  less  than  one  second, 
say  by  l/n  th  of  a  second,  it  would  gain  on  the  clock 
pendulum  and  thus  fall  out  of  coincidence  with  it,  so  that 
the  bell  would  cease  to  ring  until  n  seconds  later,  when  the 
two  pendulums  would  be  in  coincidence  again.  Let  us 
suppose  that  the  time  between  these  successive  coinci- 
dences is  100  seconds,  then  we  know  that  in  this  time  the 
clock  pendulum  has  made  one  hundred  half-vibrations 
and  the  simple  pendulum  one  more,  or  101  half-vibrations. 
In  other  words,  the  simple  pendulum  has  made  101  half- 
vibrations  in  100  seconds,  hence  the  value  of  its  half- 
period  is  100/101  seconds.  If,  on  the  other  hand,  the 
simple  pendulum  had  been  observed  to  lag  behind  the 
clock  pendulum,  and  the  time  between  successive  coinci- 
dences remained  the  same,  we  would  know  that  its  half- 
period  is  100/99  seconds. 

(a)  The  simple  pendulum  used  consists  of  a  brass 
sphere  suspended  from  a  knife-edge  by  a  wire  so  that  the 
length  is  adjustable.  Adjust  the  pendulum  so  that  its 
length  is  either  greater  or  less,  by  2  or  3  cm.,  than  that  of  a 
simple  pendulum  beating  seconds.  Two  different  lengths 


28]  THE  SIMPLE  PENDULUM  13 

(in  successive  determinations)  should  be  used  such  that 
one  is  greater  and  the  other  less  than  that  of  a  pendulum 
beating  seconds.  In  getting  the  length  it  is  well  to  measure 
with  a  meter  rod  and  square  to  the  top  of  the  ball,  and 
then  to  determine  the  diameter  of  the  ball  with  the  cali- 
pers. The  length  of  the  pendulum  is  the  distance  from 
the  knife-edge  to  the  center  of  the  ball.  Start  the  ball 
swinging  in  an  arc  of  about  10  cm.,  taking  care  not  to 
give  it  a  twisting  vibratory  motion.  During  the  vibra- 
tions watch  the  hands  of  the  laboratory  clock  and  record 
the  hour,  minute  and  second  of  each  successive  coinci- 
dence between  the  simple  pendulum  and  the  clock  pendu- 
lum, up  to  ten  or  more.  If  the  bell  rings  for  more  than 
one  swing  during  each  coincidence,  take  the  mean  of  the 
times  of  the  first  and  last  rings  as  the  time  of  the  coin- 
cidence. 

(6)  To  obtain  from  the  data  a  more  precise  average 
value  of  the  time  between  successive  coincidences,  proceed 
as  follows:  Find  the  difference  in  time  between  the  first 
and  sixth  coincidences,  the  second  and  seventh,  and  so 
on,  and  take  the  mean.  From  this  the  average  time  be- 
tween successive  coincidences  may  be  found  and  the 
period  calculated.  Be  sure  to  note  whether  the  pendulum 
was  gaining  or  losing  on  the  clock. 


(c)  Calculate  the  value  of  g  for  the  two  cases  and  take 
the  mean. 

(d)  What  effect  would  be  produced  upon  the  vibration 
of  a  pendulum  by  carrying  it,  (1)  to  a  mountain  top,  (2) 
from  the  equator  to  the  pole  of  the  earth?     In  what  way 
does  the  pendulum  used  in  this  experiment  fall  short  of 
the  requirements  for  a  simple  pendulum?     What  is  the 
object  of  taking  a  small  amplitude  of  vibration? 


14  MOMENTUM  [29 

29.     MOMENTUM. 

The  object  of  this  experiment  is  to  make  a  direct  test 
of  the  truth  of  the  second  law  of  motion  in  the  form  in 
which  the  law  was  stated  by  Newton,  viz. :  that  the  rate 
of  change  of  momentum  is  proportional  to  the  force 
producing  it.  Incidentally  it  also  serves  to  determine 
the  acceleration  of  gravity. 

The  apparatus  consists  of  a  brass  plate  about  fifteen 
inches  long  and  two  and  a  half  inches  wide  to  which 
is  soldered,  near  the  bottom,  a  funnel  of  sheet  brass 
with  a  small  re-entrant  funnel  at  its  mouth  and  with  a 
side  outlet.  At  the  upper  end  of  the  vane  are  two  project- 
ing knife  edges. 

(a)  Weigh   the   vane.      Locate  its   center   of   gravity 
and   measure   the   distance   from   the   line   of   the   knife 
edges   to  the   center  of  gravity,   the   distance  from  the 
line  of  the  knife  edges  to  the  center  of  the  funnel. 

(b)  Measure    the    diameter    of    the    nozzle    with    the 
comparator. 

(c)  Adjust  the  vane  so  that  it  swings  freely  about 
a  horizontal  axis  and  direct  a  small  swift  stream  of  water 
from  the  nozzle  horizontally  into  the  funnel.     Arrange 
a  scale  parallel  to  the  stream  and  near  one  side  of  the 
vane.     Measure  its  distance  from  the  knife  edge. 

When  conditions  are  steady,  make  five  determinations 
of  the  weight  of  water  which  is  discharged  in  measured 
time  intervals,  noting  in  each  case  the  deflection  of  the 
vane. 


(d)     From  the  diameter  of  the  nozzle  and  the  quantity 
of    water    discharged    in    a   second    calculate    the   initial 


28-29]  MOMENTUM  15 

speed  of  the  water.     Assume  that  the  final  speed  is  zero 
and  calculate  the  change  of  momentum  per  second. 

The  deflecting  force  moment  is  that  due  to  the  impact 
of  the  water  jet,  the  restoring  force  moment  is  that  due 
to  the  weight  of  the  vane.  When  the  vane  is  in  equilibrium 
these  two  moments  are  equal. 

Let  the  mass  of  the  vane  be  M,  the  angle  of  deflection 
a,  the  force  of  the  water  F,  the  distance  from  the  line 
of  the  knife  edges  to  the  center  of  gravity  /,  the  distance 
from  the  line  of  the  knife  edges  to  the  center  of  the 
funnel  L,  then 

F  L  cos  a  =  M  sin  a,  and  F  =  M  I  tan  a/  L 
Using  this  expression  calculate  the  force  F  in  grams. 

Find  the  ratio  of  the  rate  of  change  of  momentum  to 
the  force  in  each  case  and  determine  its  mean  value. 

The  constancy  of  this  ratio  affords  a  verification  of 
the  law:  and  the  mean  value  measures  the  acceleration 
of  gravity. 

30.     MOMENT  OF  INERTIA. 

A  heavy  disk  is  mounted  so  that  it  may  be  set  in  motion 
by  a  mass,  m,  suspended  by  a  cord  wound  on  its  cir- 
cumference. An  electrically  driven  tuning  fork  presses 
a  stylus  against  one  side  which  has  been  smoked  in  a 
gum  camphor  flame.  The  opposite  side  has  a  circular 
scale  graduated  in  degrees  and  two  microscopes  are 
mounted,  one  on  either  side,  to  facilitate  angular 
measurements. 

The  moment  of  inertia  of  a  body  is  defined  as  7  =  w1r12. 
For  a  solid  disk  this  may  be  shown  by  elementary  calculus 


16  MOMENT  OF  INERTIA  [30 

to  be  M  R2/  2,  when  M  is  the  mass  and  R  the  radius  of 
the  disk. 

When  torque  L  =  M  g  R  acts  on  the  disk,  an  angular 
acceleration  a.  is  produced  such  that 

L-Ia  (1) 

This  may  be  written  in  the  form  I  =  L  /  a  (constant)  (2) 
If  the  torque  L  acts  for  time  t,  we  may  write 

Lt  =  I  at  =  lu  (3) 

where  co  =  a.  Ms  the  final  angular  velocity  of  the  disk, 
supposing  it  to  have  started  from  rest. 

This  may  be  written     /  =  L  t  /  co     (constant)  (4) 

If  6  is  the  angle  turned  through  in  time  /  starting  from 
rest,  6  =  t  co/2.  Multiply  each  side  of  equation  (3)  by 
this  value  and  we  obtain,  on  cancelling  t, 

L  6  =  /  coV2  (5) 

The  product  L  6  represents  the  work  done  by  the  torque 
in  turning  through  the  angle  6,  expressed  in  radians. 
The  work  may  also  be  represented  by  m  g  s  where  5  is 
the  distance  the  mass,  m,  falls,  in  giving  the  disk  its 
velocity  co.  Equation  (5)  may  be  written  in  the  form 

/  =  2L  6  I  co2     (constant)  (6) 

If  the  weight  falls  for  exactly  one  turn,   6  =  2  TT. 

(a)  Set  the  disk  so  that  one  edge  of  it  projects  past 
the  edge  of  the  table.  Smoke  one  side  of  the  disk.  Tie 
a  thread  to  the  50  gm.  wt.  of  just  sufficient  length  that 
it  does  not  quite  slip  off  the  peg  as  the  weight  touches 
the  floor.  Wind  the  thread  up  exactly  one  full  turn 
(one  circumference  of  the  disk)  as  shown  by  the  micro- 
scope and  set  the  brake.  Connect  the  dry  cells  to  the 


30]  MOMENT  OF  INERTIA  17 

two  binding  posts  on  the  tuning  fork  and,  after  starting 
it,  adjust  the  nut  on  the  end  of  the  axis  till  the  stylus 
presses  tightly  against  the  disk.  Release  the  brake  and 
turn  the  handle  on  the  rack  only  slowly  so  that  the 
trace  on  the  smoked  surface  will  be  a  spiral  with  the 
successive  turns  pretty  close  together.  Allow  the  disk 
to  turn  at  least  two  full  turns  after  the  weight  strikes 
the  floor  before  applying  the  brake.  The  negative 
acceleration  due  to  friction  is  to  be  obtained  from  this 
part  of  the  trace. 

(6)  Take  the  disk  from  its  bearings  and  mark  with 
the  pin  the  first  distinguishable  wave  crest  and  number 
it  0.  Mark  the  10th,  20th,  40th,  80th,  120,  etc.,  crests. 
The  falling  weight  should  have  touched  the  floor  at  the 
end  of  exactly  one  turn.  Slip  ten  or  more  waves  on 
each  side  of  this  point  and  begin  again  marking  the  Oth, 

40th,  80th,  etc.,  wave  crest. 

» 

(c]  Replace  the  disk  in  its  bearings  and  set  the  two 
microscopes.      Read  the   angular   positions   of   the   start 
and  of  the  interval  marks  and  record.    If  n  is  the  frequency 
of  the  tuning  fork,  40  /  n  sec.  is  the  time  interval  between 
two    successive    marks.      The    difference    of    successive 
readings   gives   the   angular   distance   in    degrees   passed 
over  in  40  /  n  sec.     The   difference  of  these  successive 
angular    distances    gives    the    angular    acceleration    in 
degrees  per  40  /  n  sec.  per  40  /  n  sec.     In  like  manner 
determine  the  negative  acceleration  due  to  friction  and 
correct  the  previous  value.     Express  the  acceleration  in 
radians  /  sec.  /  sec. 

(d)  Repeat  (a)  using  a  100  gm.  suspended  weight. 


(e)  Plot  a  curve  for  each  trial  using  vibrations  (0, 
40,  80,  etc.)  as  ordinates  and  the  corresponding  angular 
position  readings  as  abscissae.  Prolong  the  curve  to 


18  MECHANICAL  EQUIVALENT  OF  HEAT         [30-31 

show  the  vibration  numbers  corresponding  to  the  start- 
ing point  of  the  trace,  and  360°  later.  Read  from  the 
curve  the .  total  number  of  vibrations  made  while  the 
disk  was  turning  through  the  first  360°.  This  gives  the 
time  interval  t.  Determine  the  angular  velocity  o>  from 
the  angular  distance  passed  over  in  several  40  wave 
intervals  after  the  weight  had  struck  the  floor. 

(/)  Substitute  the  known  values  for  each  trial  in 
formulae  (2),  (4)  and  (6)  and  compare  the  value  of  / 
found  with  the  value  of  MR2/  2.  Compare  m  g  s  with 
I  o>»  /  2. 

Question:  In  falling,  all  the  potential  energy  of  the 
falling  mass  does  not  go  into  kinetic  energy  of  rotation 
of  the  disk  but  some  remains  as  kinetic  energy  of  the 
falling  mass  (mv2/2).  This  was  ignored  in  the  ex- 
periment. Are  the  consequent  experimental  values  of  / 
too  large  or  too  small? 

31.       MECHANICAL   EQUIVALENT   OF   HEAT 

To  determine  the  mechanical  equivalent  of  heat  by 
Callendar's  method. 

The  number  of  units  of  mechanical  work  which  is  equiv- 
alent to  the  calorie  of  heat  is  called  the  mechanical  equiva- 
lent of  heat.  Most  of  the  methods  employed  in  determin- 
ing it  produce  the  heat  by  means  of  mechanical  work 
done  against  friction.  In  Callendar's  method  a  measur- 
able amount  of  work  done  against  the  friction  between  a 
stationary  silk  belt  and  a  revolving  vessel  is  converted 
into  heat  in  a  known  mass  of  water  contained  in  the 
vessel.  The  apparatus  consists  of  a  brass  cylindrical 
vessel  which  contains  a  known  mass  of  water  and  whose 
axis  is  horizontal.  This  cylinder  can  be  rotated  at  a 
moderate  speed  by  hand  or  by  motor.  Over  the  surface 


31]  MECHANICAL  EQUIVALENT  OF  HEAT  19 

of  the  cylinder  a  silk  belt  is  wound  so  as  to  make  one  and  a 
half  complete  turns.  From  the  ends  of  this  belt  are  sus- 
pended known  masses,  adjusted  so  as  to  provide  a  force- 
moment  which  will  oppose  the  rotation  of  the  vessel.  An 
automatic  adjustment  for  equilibrium  is  secured  by  the 
use  of  a  light  spring  balance  which  acts  in  direct  opposition 
to  the  weight  at  the  lighter  end  of  the  belt.  This  spring 
balance  contributes  only  a  small  part  to  the  effective 
difference  of  load  between  the  two  ends  of  the  belt,  hence 
small  errors  in  its  reading  are  relatively  unimportant. 
The  masses  suspended  from  the  belt  are  approximately 
adjusted  by  trial  to  suit  the  friction  of  the  belt,  the  final 
adjustment  being  automatically  effected  by  the  spring 
balance.  A  counter  registers  the  number  of  revolutions; 
and  a  bent  thermometer,  inserted  through  a  central  open- 
ing in  the  front  end  of  the  cylinder,  measures  the  tem- 
perature. 

If  M  is  the  mass  at  the  heavier  end  of  the  belt,  m  the 
mass  at  the  lighter  end,  and  F  the  reading  of  the  spring 
balance,  then  the  force  acting  to  oppose  the  rotation  of 
the  cylinder  is  (M  -  m  +  F)  g,  where  g  is  the  acceleration 
due  to  gravity.  The  work  done  in  overcoming  this  force 
during  one  revolution  of  the  cylinder  is  2wr  (M  -  m  +  F)  g. 
If,  in  n  revolutions,  the  water  of  mass  W  is  raised  from 
TV* C.  to  r2°C.,  we  have,  by  equating  the  work  done  and 
the  heat  generated: 

(1)     2irrn  (M  -  m  +  F)  g  =  (W  +  w)  (T*  -  7\  +  R)  J, 

where  w  is  the  water-equivalent  of  the  cylinder  and  the 
thermometer,  .R  is  a  temperature-correction  to  compensate 
for  radiation,  conduction,  and  the  viscosity  of  the  water, 
and  J  is  the  mechanical  equivalent  of  heat. 

(a)  Fill  the  cylinder  half  full  of  water  at  room  tem- 
perature. Do  not  use  the  bent  thermometer  to  determine 


20  MECHANICAL  EQUIVALENT  OF  HEAT  [31 

the  room  temperature,  as  bending  may  have  shifted  its 
true  scale  above  the  bend.  Suspend  masses  from  the 
ends  of  the  belt,  so  that,  when  the  cylinder  is  rotated 
at  moderate  speed,  the  masses  hang  free. 

Read  the  temperature  t\  of  the  water,  loosen  the  belt 
so  as  to  eliminate  the  friction  between  it  and  the  cylinder, 
and  give  100  turns  of  the  cylinder,  at  the  rate  just  deter- 
mined. Record  the  final  temperature  £2.  This  is  done  to 
determine  the  rate  at  which  the  temperature  of  the  water 
is  changing,  due  to  radiation,  conduction,  and  the  vis- 
cosity of  the  water,  just  before  the  run  in  (b)  is  made. 

(b)  Adjust  the  belt  and  masses  as  at  first,  read  the 
temperature  Ti  of  the  water  and  rotate  the  cylinder  at 
uniform  speed  until  the  temperature  has  risen  about  3  or 
4  degrees.    Again  record  the  temperature  T2.     Record  the 
number  of  revolutions  n,  the  masses  M  and  m,  and  the 
force  F. 

(c)  Loosen  the  belt  and  give  100  turns  at  about  the 
the  same  rate  as  used  in   (b),  recording  the  initial  and 
final  temperatures  t3  and  i*.     This  is  done  to  determine 
the  rate  at  which  the  temperature  is  changing,  due  to 
radiation,   conduction,    and  the   viscosity   of   the   water, 
just  after  the  run  in  (b)  is  made. 


(d)  For  the  n  revolutions  of  (b),  the  loss  in  temper- 
ature  due   to   radiation,    conduction,    and   the   viscosity 
of  the  water  is 

(2)  R=^L    (*.-*.  +  *.-*«)• 

100  2 

From  the  data  and  equations  (1)  and  (2)  determine  the 
value  of  /. 

(e)  Point  out  the  principal  sources  of  error. 


32]  EFFLUX  OF  GASES  21 

32.     EFFLUX  OF  GASES.     RELATIVE  DENSITIES* 

The  object  of  this  experiment  is  to  find  the  relative  densities 
of  certain  gases  from  the  observation  of  the  relative  times  of 
efflux  of  equal  volumes  of  these  gases  through  a  small  aperture. 

The  ratio  of  the  densities  of  two  gases,  under  the  same 
conditions  as  to  pressure,  is  equal,  very  approximately, 
to  the  inverse  ratio  of  the  squares  of  the  speeds  with 
which  the  gases  escape  through  a  fine  openijig  in  a  dia- 
phragm. Since  the  time  of  escape  of  a  given  volume  will 
be  inversely  as  the  speed  of  efflux,  it  follows  that  the  ratio 
of  the  densities  of  two  gases  is  equal  to  the  direct  ratio  of 
the  squares  of  the  time  of  efflux  of  equal  volumes  under  the 
same  conditions.  This  relation  was  experimentally  dis- 
covered by  Bunsen.  For  a  proof  of  it,  from  the  energy 
relations,  see  Duff,  Sect.  230. 

The  gas-holder  consists  of  a  glass  bulb,  at  the  top  of 
which  is  a  three-way  stop-cock  and  a  diaphragm  with  a 
fine  opening.  The  bulb  is  connected  with  a  reservoir  of 
oil  of  low  vapor  pressure.  The  three-way  cock  allows 
communication  to  be  made  with  the  outside  for  filling  or 
with  the  diaphragm. 

(a)  First  fill  the  bulb  with  dry  air.  To  do  this,  turn 
the  stop-cock  so  as  to  put  the  bulb  in  communication 
with  the  air,  and  allow  the  oil  to  fill  the  bulb.  This 
drives  out  most  of  the  contained  gas.  Connect  the  bulb 
with  the  calcium  chloride  drying  tube  and  raise  the 
plunger  in  the  reservoir.  This  operation  will  fill  the  bulb, 
and  by  repeatedly  emptying  and  filling  the  bulb,  it  will 
become  practically  freed  of  the  moist  air  or  other  gas 
previously  contained  in  it. 


22  EFFLUX  OF  GASES  [32 

Turning  the  stop-cock  so  that  the  gas  in  the  bulb  is 
in  communication  with  the  diaphragm,  note  the  time 
when  the  surface  of  the  oil  is  on  a  level  with  the  lower 
mark  on  the  bulb.  Again  note  the  time  when  the  oil 
passes  the  upper  mark  on  the  bulb  and  quickly  close  the 
stop-cock. 

Repeat,  making  two  or  three  determinations  of 
the  time  of  efflux  for  the  given  volume  of  air,  and  take 
the  mean. 

(6)  Repeat  (a),  filling  the  bulb  with  illuminating  gas, 
following  the  directions  there  given  for  filling  the  bulb, 
the  bulb  being  connected  directly  to  the  source  of  the 
gas  used.  Note  the  time  of  efflux  between  the  same 
two  positions  used  in  (a).  This  insures  the  same  con- 
ditions as  to  pressure  in  the  two  cases. 

(c)     Repeat  the  experiment,  using  dry  carbon  dioxide. 


(d)  Calculate  the  relative  densities,  referred  to  air,  of 
the  gas  used  in  (6).  Taking  the  density  of  dry  air  under 
standard  conditions  to  be  0.001293  gm.  per  cc.,  find  the 
density,  under  standard  conditions,  of  the  gases  used. 
What  laws  have  been  used,  or  assumptions  made,  in 
answering  the  requirement  of  the  preceding  sentence? 


33]  CALIBRATION  OF  A  SET  OF  WEIGHTS  23 

33.     CALIBRATION  OF  A  SET  OF  WEIGHTS 

By  the  calibration  of  a  set  of  weights  is  meant  the 
determination  of  the  amount  by  which  each  one  of  them 
is  in  error.  The  simplest  way  of  doing  this  is  by  com- 
paring each  weight  with  a  standard  of  the  same  nominal 
(marked)  value.  When  no  set  of  standards  is  available, 
it  is  sufficient  to  know  the  variation  within  a  given  set 
of  weights  from  their  marked  values.  This  method  of 
calibration*  is  based  on  the  assumption  that  one  of  the 
weights  of  the  set  is  correct.  The  errors  in  the  others 
are  then  determined  by  the  following  proceedure: 

Assuming  provisionally  that  one  of  the  smaller  weights 
of  the  set  is  correct,  we  determine  the  errors  in  all  the 
other  weights  using  this  as  our  standard.  When  the 
error  in  the  largest  weight  is  known  in  terms  of  this 
standard,  we  reverse  matters  and  adopt  the  largest 
weight  as  a  permanent  standard,  and  recalculate  the 
errors  on  this  basis.  The  largest  weight  is  adopted  as 
the  permanent  standard  because  it  is  least  likely  to  be 
in  error. 

(a)  The  comparisons  are  best  carried  out  by  the 
method  of  substitution.  Place  the  1'  gram  piece  on  the 
left-hand  pan  and  balance  it  approximately  by  means 
of  the  rough  unadjusted  1  gram  weight.  Take  two  sets 
of  observations  for  the  determination  of  the  rest  point 
by  the  method  of  vibrations,  and  determine  the  sensitive- 
ness of  the  balance  for  this  load,  using  the  rider,  if  necess- 
sary,  to  secure  a  rest  point  which  is  on  the  scale. 

Now  replace  the  1'  gram  piece  by  the  1"  gram  piece  to 
be  compared  with  it.  The  two  are  distinguished  by 


*Richards,  The  Jour,  of  the  Am.  Chem.  Soc.,  vol.  22,   1900. 


24  CALIBRATION  OF  A  SET  OF  WEIGHTS  [33 

small  dots  punched  above  and  to  the  right  of  figure  1. 
Do  not  change  the  load  in  the  right-hand  pan   (tare). 

Take  readings  for  a  new  determination  of  the  rest  point. 
Take  similar  readings  for  the  1'"  gram  piece. 

(6)  Place  the  2  gram  piece  on  the  left-hand  pan  and 
balance  it  approximately  by  means  of  the  unadjusted 
2  gram  weight  (tare).  Make  two  determinations  of  the 
rest  point,  and  then,  replacing  the  2  gram  weight  with 
the  1'  and  I"  gram  weights,  make  two  further  deter- 
minations. 

(c)  Proceeding  in  this  way,  compare  the  5  gram  piece 
with  the  sum  of  the  2 '  +  I'  +  I"  +  I'",  and  the  10'  gram 
piece  with  the  sum  of  the  5+2  +  1'  +  1"  +  1'",  deter- 
mining the  sensitiveness  with  each  load. 


(d)  From  the  rest  point  and  sensibility  observations 
in  (a),  determine  the  values  of  the  1"  gram  and  V"  gram 
pieces  in  terms  of  the  V  as  a  standard,  remembering 
that  an  increase  in  weight  shifts  the  rest  point  towards 
the  larger  numbers. 

Using  the  corrected  value  of  the  1"  gram  piece,  deter- 
mine from  your  data  taken  in  (b)  the  value  of  the  2  gram 
piece  in  terms  of  your  provisional  standard.  Using  these 
corrected  values,  determine  in  a  similar  fashion  the  value 
of  the  5  and  10'  gram  pieces. 

Assuming  that  the  10'  gram  piece  is  to  be  your  standard, 
find  the  proportional  part  of  its  value  which  should  be 
assigned  to  the  5,  2,  and  1  gram  pieces.  Tabulate  your 
data  and  .results  as  follows : 

Place  in  the  first  column  the  nominal  (marked)  values 
of  the  weights;  in  the  second  column,  the  preliminary 


33] 


CALIBRATION  OF  A  SET  OF  WEIGHTS 


25 


values  in  terms  of  the  1'  gram  standard.  Place  in  the 
third  column  the  proportional  part  of  the  value  of  the 
10'  gram  weight  which  should  be  assigned  to  each  weight 
on  the  assumption  that  the  10'  gram  weight  is  your 
new  standard.  By  subtracting  the  values  in  the  third 
column  from  those  in  the  second  we  obtain  the  corrections 
sought,  which  should  be  listed  in  column  four.  The 
values  in  grams  with  the  10'  gram  weight  as  the  standard 
should  be  listed  in  column  five.  A  sample  tabulation  is 
given  below. 

Calibration  of  Weights,  Set  C-7,  December,  1915,  G.  W.  R. 


1 1 

63 

O  efl 

z> 


rt   en  *c     w 

Z  %  §  6 


PH 


PU  00  f») 


.- 


§a! 


(10 

Standard 

.99984 

+  .00016 

1.00016 

d'O 

1.00036 

.99984 

+  .00052 

1.00052 

(!'") 

1.00054 

.99984 

+  .00070 

1.00070 

(2) 

2.00035 

1.99968 

+  .00067 

2.00067 

(5) 

4.99914 

4.99920 

-  .00006 

4.99994 

(ioo 

9.99839 

9.99839 

Standard 

10.00000 

26  CALIBRATION  OF  THE  TUBE  [34 

34.     ABSOLUTE  CALIBRATION    OF  A  THERMO- 
METER 

To  plot  a  curve  from  which  the  true  temperature  may  be 
obtained  corresponding  to  each  scale-reading  of  a  given 
mercurial  thermometer. 

Such  a  curve  is  called  the  calibration  curve  of  the  ther- 
mometer. The  process  of  obtaining  it  is  absolute  since 
it  does  not  involve  comparison  with  a  standard  ther- 
mometer. This  process  consists,  first,  in  determining 
the  absolute  corrections  for  two  separate  scale-readings 
on  the  thermometer,  preferably  near  its  fixed  points, 
that  is,  the  points  on  its  scale  corresponding  to  the  tem- 
perature of  melting  ice  and  the  temperature  of  water 
boiling  under  standard  pressure.  The  thermometer  tube 
between  the  two  points  is  then  calibrated  because  of  the 
possibility  that  its  bore  may  not  be  uniform,  and  the 
relative  corrections  thus  determined  are  superimposed 
graphically  upon  the  correction  curve  resulting  from 
the  absolute  corrections  at  or  near  the  two  fixed  points. 

(a)  Calibration  of  the  Tube. — Break  off  a  portion 
of  the  thread  of  mercury  about  ten  degrees  in  length. 
For  this  purpose,  first  invert  the  thermometer  and  let 
enough  mercury  flow  into  the  small  cistern  to  fill  it  about 
half  full;  then,  holding  the  thermometer  in  a  horizontal 
position,  tap  or  jar  it  lightly  lengthways  to  break  the 
mercury  in  the  small  cistern  loose  from  the  rest.  The 
•mercury  in  the  stem  will  now  flow  back  into  the  bulb 
and  leave  the  stem  free  for  the  thread  of  mercury  which 
must  be  jarred  loose  from  the  mercury  in  the  small  cis- 
tern. Ask  for  assistance,  if  necessary. 

Jar  the  lower  end  of  the  thread  to  the  zero-point  of 
the  scale  and  read  the  position  of  the  upper  end  (which 


34]  CORRECTION  FOR  THE  FIXED  POINTS         27 

will  be  near  the  10°  mark)  to  tenths  of  a  degree.  Then 
jar  the  lower  end  of  the  thread  to  the  10°  mark  and  read 
the  upper  end.  Repeat  with  the  lower  end  at  the  succes- 
sive points  20°,  30°,  40°,  etc.,  up  to  90°.  Then  take  the 
readings  in  the  reverse  order,  setting  the  upper  end  of 
the  thread  successively  at  100°,  90°,  etc.,  down  to  10°, 
and  reading  the  position  of  the  lower  end  each  time. 
The  object  of  these  readings  is  to  find  the  length  of  the 
thread  in  each  of  the  ten  intervals  between  0°  and  100° ; 
by  means  of  the  two  series  it  is  possible  to  find  the 
average  value  of  the  length  of  the  thread  for  each  interval. 

(6)     Correction  at  the  Lower  Fixed  Point.— Put 

the  thermometer  through  the  cork  in  a  test-tube,  having 
filled  the  latter  about  half  full  of  distilled  water.  Place 
the  tube  in  a  freezing  mixture  of  shaved  ice  and  salt, 
and  stir  the  water  around  the  thermometer  until  it  begins 
to  freeze.  Read  the  thermometer.  By  warming  the  tube 
in  the  hand  and  repeating  the  freezing  process,  obtain 
several  readings. 

(c)  Correction  Near  the  Upper  Fixed  Point.— 

Place  the  thermometer  through  the  cork  in  the  tube  at 
the  top  of  the  boiler,  with  the  bulb  well  above  the  sur- 
face of  the  water.  Boil  the  water  so  that  the  steam  issues 
freely,  but  not  with  any  perceptible  pressure,  from  the 
vent.  Read  the  thermometer  when  it  becomes  steady. 
Allow  the  boiler  to  cool  slightly,  and  repeat,  making 
three  readings  in  all.  If  the  instrument  be  provided 
with  a  water-manometer,  take  the  manometer-reading 
simultaneously  with  the  temperature-reading.  Read  the 
barometer. 

(d)  Let  the  thermometer  cool  slowly  to  about  the  tem- 
perature of  the  room,   and  repeat   (6).     If  the  freezing 
point  observed  now  is  different  from  that  observed  in 


28 


CORRECTION  FOR  THE  FIXED   POINTS 


[34 


(b),  use  the  mean  of  the  two  values  in  the  calibration 
that  follows. 


(e)  Assume  that  the  temperature  of  freezing  water 
is  0°C.  From  the  Tables  take  the  true  boiling-point 
temperature  for  the  pressure  observed  in  (c),  find  the 
corrections  of  the  thermometer  for  the  scale-readings 
observed  in  (b)  and  (c).  Record  these  two  corrections 
by  points  on  coordinate  paper,  having  as  abscissae  the 


-04 


Fig-  2. — This  plot  shows  the  calibration  curve  corresponding  to  the 
sample  data  given  on  page  18,  with  an  observed  freezing  point  of 
+  0.2°C  and  boiling  point  of  99.4°C  when  the  true  temperature  was 
99.8°C. 

scale-readings  of  the  given  thermometer  in  degrees,  and 
as  ordinates  the  corresponding  corrections  in  tenths  of 
a  degree  but  on  a  magnified  scale.  (See  Fig.  2).  Correc- 
tions should  be  plus  (+)  if  they  are  to  be  added  to  the 
observed  to  give  the  true  temperatures,  minus  (.— )  if 
they  are  to  be  subtracted. 


34]  THERMOMETER  CALIBRATION  29 

Connect  these  two  points  by  a  straight  line.  The 
ordinate  of  this  straight  line  at  any  point  gives  the  cor- 
rection of  the  thermometer  at  that  scale-reading  on  the 
assumption  that  the  bore  of  the  thermometer  is  uniform 
throughout  the  whole  range.  In  general  this  assump- 
tion is  not  justified,  and  there  must  be  added  to  this 
correction  at  each  point  another  correction  due  to  the 
inequalities  of  the  diameter  of  the  bore. 

In  order  to  find  the  bore  correction,  proceed  as  follows. 
Determine  the  mean  length  of  the  thread  in  all  of  its 
different  positions.  If  the  bore  were  uniform,  each  of 
the  observed  lengths  would  equal  this  mean  value.  From 
the  deviation  between  the  observed  and  mean  thread 
lengths,  calculate  the  length  which  a  thread  whose  mean 
length  was  exactly  ten  degrees  would  have  when  placed 
in  the  first  interval.  This  would  be  in  the  same  ratio  to 
the  observed  thread  length  as  ten  degrees  is  to  the  mean 
thread  length.  We  will  call  this  the  reduced  thread  length. 
An  actual  rise  of  exactly  ten  degrees  in  temperature  will 
produce  an  apparent  rise  equal  to  this  amount.  The 
difference  between  this  reduced  thread  length  and  ten 
degrees  when  the  sign  is  reversed,  gives  the  bore  correction 
for  a  temperature  of  approximately  ten  degrees. 

Proceed  similarly  with  each  interval.  The  total  cor- 
rection which  must  be  applied  at  any  point  of  the  scale 
is  found,  by  adding  together  the  corrections  for  all  the 
intervals  below  that  point. 

Since,  by  definition,  the  mean  thread  length  is  one 
tenth  the  sum  of  all  the  observed  thread  lengths,  the 
total  bore  correction  at  100°  will  necessarily  be  zero. 

The  sample  set  of  data  given  below  will  illustrate  the 
method.  /  is  the  observed  thread  length  in  a  given  interval, 


so  THERMOMETER  CALIBRATION  [34 

which  may  be  an  average  of  several  determinations.  L  is 
its  mean  value  for  all  intervals;  r  is  the  reduced  thread 
length  in  a  given  interval.  Its  mean  value  is  ten  degrees. 
10  -  r  gives  the  correction  for  each  interval,  and  the  sum 
of  all  such  corrections  gives  the  total  at  a  certain  point 
of  the  scale. 

Intervals         Average/  r          Dif.(10— r)  Bore- Corrections 

1st                   14.16°      10.01°        -.01°            At  0°=   .00° 

2nd                 14.25°      10.07°        -.07°              "  10°  =-.01° 

"  20°  =-.08° 

"  30°  =-.17° 

"  40°  =-.36° 

11  50°  =-.45° 

"  60°  =-.44° 

"  70°  =-.43° 

"  80°  =-.32° 

"  90°  =-.21° 

"  100°=   .00° 

The  curve  showing  the  resultant  corrections  for  all 
scale-readings  from  0  to  100  can  therefore  be  obtained  by 
super-imposing  the  bore-corrections  just  found  upon  the 
line  drawn  to  show  the  corrections  due  to  the  errors  in 
the  fixed  points.  For  this  purpose  plot  points  whose 
abscissae  are  10°,  20°,  30°,  etc.,  and  whose  corresponding 
ordinates  are  found  by  measuring  from  the  slanting  line, 
already  drawn,  distances  equal  to  the  corresponding  bore- 
corrections — measuring  up  or  down  from  this  line  accord- 
ing as  the  corrections  are  plus  or  minus.  The  smooth 
curve,  which  should  now  be  drawn  through  these  plotted 
points,  is  the  calibration  curve  of  the  thermometer. 

What  are  the  true  temperatures  corresponding  to  the 
scale-readings  0°,  25°,  50°,  75°  and  100°  on  the  given 
thermometer? 


3rd 

14.27° 

10.09° 

-.09° 

4th 

14.42° 

10.19° 

-.19° 

5th 

14.27° 

10.09° 

-.09° 

6th 

14.13° 

9.99° 

+.01° 

7th 

14.13° 

9.99° 

+.01° 

8th    . 

14.00° 

9.89° 

+.11° 

9th 

14.00° 

9.89° 

+.11° 

10th 

13.85° 

9.79° 

+.21° 

Mean 

=  14.15°  =L 

10.00° 

35]  HEAT  OF  SOLUTION  31 

35.     HEAT  OF  SOLUTION. 

To  find  the  quantity  of  heat  liberated  by  the  solution  of 
one  gram  of  a  given  salt  in  water. 

When  a  salt  is  dissolved,  the  temperature  may  rise 
or  fall,  due  to  the  liberation  or  absorption  of  heat  energy. 
In  the  latter  case,  the  heat  of  solution  is  said  to  be  nega- 
tive. The  amount  of  heat  evolved  depends  on  the  final 
concentration  of  the  solution,  being  greatest  when  the 
solution  is  carried  to  infinite  dilution. 

The  quantity  of  heat  evolved  is  measured  in  terms  of 
the  change  in  temperature  of  the  solution  itself.  Now 
the  specific  heat  of  a  solution  may  differ  very  appreciably 
from  the  specific  heat  of  water,  and  can  only  be  accurately 
determined  by  experiment.  Approximately,  however, 
the  heat  capacity  of  a  solution  is  the  same  as  though 
it  consisted  of  a  simple  mixture  of  water  and  the  dissolved 
material,  in  the  solid  form.  We  will  base  our  calculations 
on  the  assumption  that  error  introduced  by  this  approx- 
imation is  negligible. 

The  following  thermal  changes  occur  during  solution: 
(1)  Potential  energy  is  transformed  into  heat  (positive 
heat  of  solution)  or  heat  into  potential  energy  (negative 
heat  of  solution).  (2)  Heat  is  absorbed  (or  given  up) 
by  the  water,  calorimeter,  etc.,  in  changing  from  the 
initial  to  the  final  temperature.  (3)  Heat  is  absorbed 
(or  given  up)  by  the  salt  in  changing  from  the  room 
temperature  to  the  final  temperature  reached  by  the 
solution. 

Calling  M  the  mass  of  water  used,  W  the  water- 
equivalent  of  the  calorimeter  and  accessories,  w  the 
water-equivalent  of  the  solid  salt,  to  the  room  temperature, 


32  HEAT  OF  SOLUTION  [35 

ti  and  tz  the  initial  and  final  temperatures  respectively 
of  the  water,  and  5C  the  heat  evolved  per  gram  of  solute 
on  forming  a  solution  of  concentration  c,  write  the 
equation  which  will  enable  you  to  calculate  5C. 

(a)  Fill  the  calorimeter  about  two-thirds  full  of  water 
and,  by  weighing,  determine  the  amount  of  water  used. 
Weigh  out  a  portion  of  the  salt  furnished,  sufficient  to 
give  a  20  per  cent  solution  (weight  of  salt  20  per  cent 
of  the  total.) 

(b)  The  stopper  and  thermometer  should  fit  tightly 
enough  so  that  the  whole  calorimeter  may  be  shaken  to 
bring  about  uniform  temperature  in  the  water  or  solution. 
Shake  until  the  water  has  reached  a  constant  temperature. 
Throw  in  the  salt,   and  after  further  shaking  read  the 
temperature  when  it  has  again  become  constant.     Record 
the  room  temperature. 

(c)  Repeat  (a)  and  (b),  using  portions  of  salt  giving 
concentrations  of  15,  10,  and  5  per  cent,  approximately. 

(d)  Determine  the  water-eqiuvalent  of  the  calorimeter 
and   accessories   by  the   method   explained   on   page   44, 
Part  1   (Calorimetry). 


(e)  Calculate  the  heat  evolved  per  gram  of  solute 
in  each  of  the  three  cases.  Use  the  tabular  value  of 
specific  heat  for  the  solid  salt,  and  assume  it  to  be  at 
room  temperature  before  solution. 

Draw  a  curve  showing  the  relation  between  heat 
evolved  and  concentration.  The  curve  may  be  assumed 
to  be  a  straight  line,  and  when  produced  to  the  axis  of 
zero  concentration,  gives  the  heat  of  solution. 


35-36]  HEAT  OH  NEUTRALIZATION  33 

(/)  To  estimate  the  error  of  your  result,  read  off  the 
deviation  of  each  plotted  point  from  the  curve.  The 
mean  deviation,  expressed  in  per  cent,  gives  the  approx- 
imate probable  error. 

36.     HEAT  OF  NEUTRALIZATION. 

When  an  aqueous  solution  of  a  strong  acid  is  poured 
into  an  aqueous  solution  of  a  strong  alkali  until  a  neutral 
mixture  is  formed,  the  essential  chemical  reaction  which 
occurs  is  the  formation  of  water.  For  instance,  if  aqueous 
solutions  of  hydrochloric  acid  and  sodium  hydroxide  are 
made  to  form  a  neutral  mixture,  although  the  mixture  is 
a  solution  of  sodium  chloride  (table  salt),  the  only  chemical 
reaction  occurring  is  the  formation  of  water.  The  heat 
generated  is  called  the  heat  of  neutralization.  The  object 
of  the  present  experiment  is  to  determine  the  heat  of  neutral- 
ization corresponding  to  the  formation  of  a  gram-molecular 
weight  of  water.  In  the  case  just  mentioned,  this  will 
occur  when  1000  gm.  each  of  normal  solutions  of  the  acid 
and  the  alkali  are  mixed. 

A  0.5  normal  solution  of  each  of  the  above  compounds 
is  furnished.  By  a  normal  solution  is  meant  one  which, 
in  1000  cubic  centimeters  of  the  solution,  contains  a 
mass  of  the  compound  (which  is  to  enter  into  the  new 
combination)  equal  in  grams  to  its  molecular  weight. 
Thus  the  normal  solution  of  sodium  hydroxide  is  a  solution 
which  contains,  in  1000  cc.  of  the  solution,  40  gm.  (23  + 
16  +  1)  of  sodium  hydroxide,  or  23  gm.  of  sodium,  16  gm. 
of  oxygen,  and  1  gm.  of  hydrogen.  The  0.5  normal 
solution  contains  one-half  as  much  in  -the  same  volume  of 
solution. 

It  is  evident  that  if  equal  volumes  of  these  solutions 
be  mixed,  the  reaction  will  be  just  completed,  and  the 


34  HEAT  OF  SOLUTION  [36 

result  will  be  a  neutral  solution  of  sodium  chloride.  The 
solutions  are  to  be  mixed  in  the  calorimeter  cup  at  as 
nearly  as  possible  the  same  temperature,  and  the  resulting 
rise  of  temperature  noted.  The  alkali  should  be  placed 
in  the  cup,  and  the  acid  added  to  it.  The  acid,  being 
immediately  neutralized,  will  then  have  no  action  on 
the  metal  of  the  cup. 

(a)  Measure   out    100   cc.    of   the   sodium   hydroxide 
solution  in  the  cup,  and  the  same  volume  of  the  hydro- 
chloric acid  solution  in  the  beaker.     Wet  the  inside  of 
the   beaker   with   the   acid   solution   before   pouring  the 
measured  amount  into  it.     This  is  to  compensate  for  the 
liquid  which  remains  in  the  beaker  when  later  it  is  emptied. 
A  small  error  is  introduced  by  taking  the  second  ther- 
mometer out  of  the  beaker  after  reading  its  temperature, 
but  this  may  be  neglected. 

If  care  has  been  taken  not  to  handle  the  cup  and  beaker 
any  more  than  is  necessary,  the  two  temperatures  should 
be  very  nearly  the  same  when  ready  for  use.  It  may  safely 
be  assumed  that  the  resulting  solution  of  sodium  chloride 
has  risen  to  the  final  temperature  from  the  mean  of  the 
two  initial  temperatures.  Make  two  trials. 

(b)  Repeat    the    work    with    solutions    of    potassium 
hydroxide  and  sulphuric  acid. 


Calculate  for  each  set  of  data  the  quantity  of  heat 
which  would  have  been  evolved  if  1000  cc.  of  the  normal 
solution  had  been  used  in  each  case.  Will  this  cause  the 
formation  of  one  gram-molecular  weight  of  water? 

The  specific  heat  of  the  sodium  chloride  solution  is 
0.987  calories,  of  the  potassium  sulphate  solution  0.985 
calories  per  gram. 


37]  VARIATION  OF  BOILING  POINT  35 

37.     VARIATION  OF  BOILING  POINT  WITH 
PRESSURE. 

References.— Duff,  pp.  227,  228;  Kimball,  p.  298. 

There  are  two  methods  employed  in  studying  the 
variation  in  the  boiling  point  of  a  liquid  with  the  pressure 
upon  its  free  surface.  By  the  dynamic  method  the  pressure 
above  the  boiling  liquid  is  varied  by  means  of  an  air-pump 
and  the  corresponding  temperature  observed.  By  the 
static  method  the  temperature  of  the  liquid,  suitably 
enclosed,  is  varied  by  means  of  baths  and  the  correspond- 
ing pressure  observed. 

The  object  of  the  present  experiment  is  to  study  the  variation 
in  the  boiling  point  of  water,  employing  the  dynamic  method. 

The  apparatus  consists  of  an  air-tight  boiler  to  hold  the 
liquid,  a  steam-condenser,  around  which  cold  water  cir- 
culates, an  air-tight  chamber  large  enough  to  equalize 
sudden  changes  in  the  pressure,  an  air-pump  for  reducing 
the  pressure  and  a  manometer  for  measuring  the  same. 
These  are  connected  up  in  the  order  named  and  made 
air-tight  so  far  as  the  air  outside  is  concerned. 

(a)  The  circulation  of  water  should  first  be  started 
through  the  steam-condenser,  which  is  a  glass  or  metal 
tube  to  jacket  the  tube  leading  from  the  boiling-flask, 
thus  condensing  the  steam  as  it  comes  from  the  flask.  The 
thermometer  should  be  passed  through  the  stopper  of  the 
flask  and  so  regulated  that  its  bulb  will  be  in  the  rising 
steam,  and  not  in  the  water.  The  connection  with  the 
large  glass  bottle  serves  to  equalize  sudden  changes  in 


-.36  VARIATION  OF  BOILING  POINT  [37 


pressure  due  to  irregularities  in  the  boiling.  In  heating 
the  water  do  not  play  the  flame  on  the  flask  directly  under 
the  glass  beads,  but  rather  to  one  side  and  below  the 
water-line. 

First  boil  the  water  at  atmospheric  pressure,  reading 
the  manometer  and  noting  the  temperature.  Then  take 
a  series  of  readings  at  intervals  of  about  5  cm.  pressure, 
until  the  "bumping"  becomes  so  violent  as  to  render 
further  readings  impracticable.  Before  each  reading, 
after  pumping  to  the  pressure  desired,  close  the  stop- 
cock over  the  jet-pump,  wait  a  short  time  for  the  pressure 
to  reach  equilibrium,  and  then  make  the  reading  of  boiler 
temperature  and  corresponding  pressure.  Put  the  pump 
again  in  connection,  obtain  a  new  pressure,  and  repeat 
the  readings.  Before  turning  off  the  water  at  the  jet,  be 
sure  each  time  to  let  air  into  the  apparatus  by  opening 
the  pinch-cock  nearest  the  pump,  otherwise  water  will 
flow  back  into  the  tubing. 

(b)  Take  a  series  of  readings  with  increasing  pressures 
up  to  atmospheric  pressure,  choosing  values  different 
from  the  previous  ones. 


(c)  Plot  the  observations  on  coordinate  paper,  using 
pressures    as    ordinates    and   temperatures    as   abscissae. 
From  the  curve  find  the  boiling  point  of  water  at  a  pressure 
of  i  atmosphere. 

(d)  Discuss   the   phenomena    of   this    experiment   in 
connection   with  the   difficulties   experienced  in   cooking 
food   at   high    altitudes.      Could    determinations   of   the 
boiling  point  of  water  be  used  to  measure  altitude,  and 
how? 


38]  CONSTANT  VOLUME  AIR  THERMOMETER          37 

38.     CONSTANT-VOLUME   AIR- THERMOMETER. 

References. — Duff,  p.  202. 

The  object  of  this  experiment  is  to  study  the  law  of  varia- 
tion of  the  pressure  of  a  given  mass  of  enclosed  air  whose 
volume  is  kept  constant  while  its  temperature  is  changed. 

The  air  is  enclosed  in  a  glass  bulb  mounted  on  a  frame. 
The  frame  is  placed  near  a  table  so  that  the  bulb  may  be 
surrounded  by  a  water-bath,  by  shaved  ice,  or  by  a  steam- 
bath,  the  table  and  an  iron  stand  being  made  use  of  to 
support  each  bath  in  turn.  A  thermometer  is  placed  in 
the  bath  to  give  its  temperature.  The  pressure  on  the 
enclosed  gas  is  regulated  by  raising  or  lowering  the  open 
tube.  The  value  of  this  pressure  may  be  determined 
from  the  barometer-reading  and  the  difference  in  the 
levels  of  the  mercury  on  the  two  sides  of  the  frame. 
Each^time  before  taking  the  readings,  the  volume  of  the 
air  in  the  bulb  is  made  the  same  by  bringing  the  mercury 
meniscus  to  the  level  of  the  wire  point  inside  the  glass 
tube  attached  to  the  bulb. 

Caution: — The  mercury  oh  the  bulb  side  should  always 
be  lowered  some  distance  before  changing  to  a  lower 
temperature.  Be  especially  careful  to  do  this  before  re- 
moving the  steam-bath  when  you  have  taken  a  reading  at 
the  boiling  point;  otherwise,  on  cooling,  the  mercury  will 
run  into  the  bulb.  Do  not  hurry  in  taking  the  readings 
after  changing  the  temperature,  but  wait  until  the  meniscus 
set  at  the  wire-point  remains  stationary. 

(a)  Without  any  bath  in  the  reservoir,  while  all  is  at 
the  room-temperature,  bring  the  mercury  to  the  wire 
point  and  determine  the  difference  in  level  of  the  mercury 
columns.  Record  the  room-temperature,  and  the  barom- 
eter-reading. 


38          CONSTANT  VOLUME  AIR  THERMOMETER          [38 

(b)  After  having  lowered  the  mercury  on  the  bulb 
side,  surround  the  bulb  with  shaved  ice,  and  then  deter- 
mine the  pressure  with  the  menicus  at  the  wire  point. 
The  temperature  may  be  taken  as  0°C. 

Melt  the  ice  with  warm  water,  and  then  make  a  series 
of  determinations  of  the  pressure  when  the  water  in  the 
vessel  is  successively  at  a  temperature  of  (approximately) 
10°,  20°,  30°,  45°,  60°,  and  80°C. 

Remove  the  water-bath,  substitute  a  steam-bath  in  its 
place,  and  make  another  determination.  The  temperature 
of  the  steam-bath  may  be  found  by  determining  the  boiling 
point  of  water  from  the  known  atmospheric  pressure  (see 
Tables). 

Arrange  all  observations  in  tabular  form. 


(c)  Plot  on  coordinate  paper  the  results  of  (6),  using 
temperatures  as  abscissae  and  the  corresponding  pressures 
as  ordinates.    Draw  a  smooth  curve  which  will  best  repre- 
sent the  average  position  of  the  points  of  the  plot,  and 
extend    the    curve    until    it    intersects    the    line    of    zero 
pressure. 

Calculate  from  the  curve  the  mean  increase  of  pressure 
per  degree  increase  in  temperature  from  0°C.  to  100°C., 
and  divide  the  result  by  the  pressure  at  0°C.,  using  values 
taken  from  the  plot.  This  is  the  temperature  coefficient 
(ft)  of  pressure  of  a  gas.  Write  it  as  a  decimal  and  find  its 
reciprocal.  The  negative  of  this  represents  what  point 
on  the  absolute  scale  of  temperatures? 

(d)  Write  an  equation  connecting  PQ,  the  pressure  at 

0°;  Pt,  the  pressure  at  t°\  t\  and  ft. 

t 

Using  this  equation  and  the  pressure  obtained  in  (a), 
calculate  the  temperature  of  the  room,  thus  using  the 


38-39]          VAPOR  PRESSURE  AND  TEMPERATURE          39 

apparatus  as  a  thermometer.     Compare  the  result  with 
the  room  temperature  as  read  from  a  mercury  thermometer. 

Show  from  your  results  how  the  pressure  of  the  gas 
varies  with  the  absolute  temperature,  the  volume  remain- 
ing constant. 

39.     VAPOR-PRESSURE  AND  TEMPERATURE. 

References. — Duff,  p.  222;  Kimball,  p.  296. 

The  object  of  this  experiment  is  to  study  the  relation  be- 
tween the  temperature  and  the  pressure  of  saturated  water- 
vapor. 

The  method  employed  is  that  referred  to  in  Exp.  37  as 
the  "static"  method  of  determining  the  boiling  point  of  a 
liquid  at  different  pressures.  Two  barometer  tubes, 
filled  with  mercury,  are  inverted  and  mounted  side  by 
side  in  a  vessel  of  mercury.  One  of  the  tubes  contains, 
above  the  mercury,  water-vapor  with  an  excess  of  water 
present,  while  the  other  tube  is  left  to  be  used  as  a  barom- 
eter. By  means  of  a  water-bath  surrounding  the  upper 
half  of  the  tubes,  the  temperature  of  the  water-vapor  can 
be  brought  to  any  desired  point.  The  bath  is  connected 
to  a  heater  and  the  change  in  temperature  is  brought 
about  by  circulation.  The  pressure  of  the  saturated 
water-vapor  at  any  temperature  will  be  the  difference 
between  the  heights  of  the  mercury  columns  on  the  two 
tubes. 

At  each  temperature  the  pressure  of  a  saturated  vapor 
of  a  given  liquid  has  a  definite  value  which  depends  on  the 
temperature  and  the  nature  of  the  liquid,  but  is  independ- 
ent of  the  volume  of  the  vapor.  When  the  temperature  is 
raised,  not  only  is  the  vapor  heated  and  the  pressure 
raised,  but  more  liquid  is  vaporized,  so  there  are  two 


40  VAPOR  PRESSURE  AND  TEMPERATURE  [39 

influences  tending  to  increase  the  pressure  of  the  vapor. 
The  purpose  of  the  present  experiment  is  to  plot  the  curve 
which  shows  how  rapidly  the  vapor-pressure  increases  as 
the  temperature  is  raised,  in  the  case  of  water-vapor. 

(a)  Read  the  heights  of  the  mercury  columns  in  the 
two  tubes  for  ten  different  temperatures  between  room- 
temperature    and    80°C.      Record    the    height    of    each 
column    separately.      To    raise    the    temperature    about 
5°  or  10°  at  a  time,  heat  the  boiler  only  for  two  or  three 
minutes,   then  remove  the  burner,   and   stir  the   water- 
bath    until    a    uniform    temperature    prevails.      By   this 
time  the  water-vapor  inside  the  tube  will  have  reached 
the   temperature   of   the   bath.      In   taking   the   temper- 
ature-readings   hold     the     bulb      of    the    thermometer 
slightly  below  the  center  of  the  space  filled  with  water- 
vapor. 

Always  wait  until  conditions  have  become  steady  before 
taking  readings  at  a  new  temperature. 

(b)  By  replacing  the  hot  water  in  the  bath  with  cold 
water,  a  little  at  a  time,  take  a  second  series  of  readings 
down  to  about  room  temperature. 


(c)  Plot  a  curve  from  the  results  of  (a)  and  (b),  with 
the  pressures  of  the  saturated  water-vapor  as  ordinates 
and  the  temperatures  as  abscissae.  Draw  the  curve  so 
that  it  will  represent  the  average  positions  of  all  the 
plotted  points. 

Determine  from  the  curve  the  boiling  point  of  water 
at  a  pressure  of  30  cm. 

Does  the  pressure  of  saturated  water-vapor  increase 
with  the  temperature  more  or  less  rapidly  than  does  the 
pressure  of  a  gas  kept  at  constant  volume? 

Would  the  results  be  different  if  the  volume  of  the 
saturated  vapor  were  kept  constant? 


40]  HYGROMETRY  41 

40.     HYGROMETRY. 

References. — Duff,  p.  279;  Kimball,  p.  304;  Edser,  p.  240. 

In  this  experiment  the  dew-point  and  the  relative  and 
absolute  humidity  of  the  air  are  to  be  determined. 

The  absolute  humidity,  d,  is  the  density  of  the  water- 
vapor  present  in  the  air,  and  is  usually  expressed  in  grams 
per  cubic  meter.  The  relative  humidity  is  the  ratio  of  the 
amount  of  water-vapor  actually  present  in  the  air  to  the 
amount  required  to  saturate  it  at  the  same  temperature, 
the  latter  quantity  being  the  maximum  amount  of  water- 
vapor  that  can  be  held  in  suspension  at  that  temperature. 
The  relative  humidity  is  therefore  equal  to  d/D,  where  D 
is  the  maximum  density  of  the  water-vapor  at  the  given 
temperature.  The  dew-point  is  the  temperature  at  which 
the  amount  of  water  actually  present  in  the  air  would  sat- 
urate it,  that  is,  the  temperature  to  which  the  air  must  be 
lowered  before  the  condensation  of  water  will  begin.  The 
pressure  of  water-vapor  is  the  pressure  which  it  would 
exert  by  itself  if  there  were  no  air  present  in  the  space 
considered.  By  Dalton's  law  this  is  the  pressure  it  actually 
does  exert  when  mixed  with  air. 

If  the  unsaturated  vapor  -be  assumed  to  behave  like 
a  perfect  gas,  the  ratio  of  actual  to  saturated  vapor  density 
equals  the  ratio  of  actual  to  saturated  vapor  pressure. 
Since  cooling  vapor  to  the  dew-point  does  not  change  its 
pressure,  the  saturated  pressure  at  the  dew-point  equals 
the  actual  pressure  at  atmospheric  temperature,  and 
this  may  be  read  from  the  table  without  correction.  In 
this  way  we  justify  the  ordinary  proceedure  of  determin- 
ing relative  humidity  as  the  ratio  of  the  pressure  p  of 
the  water-vapor  in  the  air  to  the  pressure  P  of  saturated 


42  HYGROMBTRY  [40 

water- vapor  at  that  temperature;  that  is,  relative  humidity 
is  equal  to  p/P. 


(I)     Alluard's   Hygrometer. 

(a)  Partially  fill  the  hygrometer  with  ether,  and 
place  a  thermometer  in  the  liquid.  Cool  the  liquid  by 
pumping  air  slowly  through  it.  When  the  tube  and  the 
air  immediately  above  it  are  cooled  to  the  dew-point, 
moisture  appears  on  the  tube,  this  being  detected  more 
easily  by  comparison  with  the  plate  on  either  side.  Note 
the  temperature  at  which  the  dew  begins  to  form.  Allow 
the  tube  to  become  warm  and  record  the  temperature  at 
which  the  dew  disappears.  Take  the  mean  of  these  two 
as  the  dew-point.  Make  three  such  determinations  of 
the  dew-point. 


(6)  From  the  Tables  find  the  pressure  of  saturated 
water-vapor  at  the  dew-point  and  also  at  the  temperature 
of  the  room,  and  calculate  the  relative  humidity.  The 
absolute  humidity  may  be  found  by  multiplying  the  rela- 
tive humidity  by  D,  the  number  of  grams  of  saturated 
water-vapor  in  a  cubic  meter  of  air  at  the  room-tem- 
perature (see  the  Tables). 

II.  Wet-  and  Dry-bulb  Hygrometer,  or  Augustus  Psy- 
chrometer. 

(a)  In  the  wet-  and  dry-bulb  hygrometer,  one  bulb  is  cov- 
ered with  wicking  which  dips  into  water,  so  that  the  bulb  is 
cooled  by  evaporation.  Swing  the  hygrometer  back  and 
forth  in  the  air  so  as  to  increase  the  circulation  of  air  about 
the  wet  bulb.  After  the  two  thermometers  come  to  con- 
stant temperatures,  record  the  temperature  t  of  the  dry 


40]  HYGROMKTRY  43 

bulb,  and  the  temperature  ti  of  the  wet  bulb.     Read  the 
barometer. 


(b)   The  following  empirical  formula  may  then  be  used : 
p  =  ^-0.00086  (t-ti), 

where  p  is  the  pressure  of  water-vapor  present  in  the 
atmosphere  and  the  value  of  which  is  to  be  found;  p\  the 
pressure  of  saturated  vapor  at  the  temperature  of  the 
wet-bulb  (obtained  from  the  Tables) ;  and  b  is  the  baro- 
metric pressure,  all  being  expressed  in  millimeters  of  mer- 
cury. Find  the  pressure  P  of  saturated  water-vapor  at 
the  room-temperature  from  the  Tables,  and  calculate  the 
relative  humidity.  Find  then  the  absolute  humidity  as 
in  (b).  From  the  Tables  and  the  readings  of  the  wet-  and 
dry-bulb  hygrometer,  find  the  dew-point. 

Compare  the  values  obtained  in  I  and  II  for  the  humid- 
ity and  the  dew-point. 


44  RATIO  OF  THE  TWO  SPECIFIC  HEATS  [41 

41.     RATIO  OF  THE  TWO  SPECIFIC  HEATS  OF 

AIR. 

References. — Duff,  p.  283;  Edser,  p.  321. 

The  object  of  this  experiment  is  to  obtain  the  value  of  the 
ratio  y  of  the  specific  heat  of  air  at  constant  pressure  to  its 
specific  heat  at  constant  volume. 

The  method  employed  is  a  modification  of  that  used 
first  by  Clement  and  Desormes.  A  quantity  of  the  gas, 
compressed  in  a  large  flask,  is  momentarily  put  in  com- 
munication with  the  atmosphere  to  allow  its  pressure  to 
fall  adiabatically  to  atmospheric  pressure,  its  temperature 
simultaneously  falling  a  little.  The  gas,  when  shut  off 
again  from  the  atmosphere,  gradually  warms  up  to  its 
initial  temperature,  causing  an  appreciable  rise  in  its 
pressure.  Let  pi  be  the  pressure  in  the  compressed  gas  at 
the  start,  Vi  the  volume  of  unit  mass  of  the  gas  and  ti  its 
temperature  (the  same  as  that  of  the  room).  Let  p0,  vt, 
and  tz  be  the  corresponding  values  of  these  quantities 
immediately  after  communication  between  the  com- 
pressed gas  and  the  atmosphere  is  established.  Then 
p2,  v2,  and  ti  will  be  the  values  of  these  same  quantities 
at  the  end,  if  p2  is  the  final  pressure.  The  gas  has  now  been 
in  three  conditions,  as  follows: 

Condition         Pressure       Vol.  of  \gm.     Temperature 
I.  pi  Vi  ti 

II.  po  V,  tt 

III.  pt  V*  h 

The  change  from  I  to  II  was  adiabatic,  since  no  time  was 
allowed  for  heat  to  pass  in  or  out  of  the  gas  by  conduction 
or  radiation;  hence,  by  the  law  for  adiabatic  changes  in  a 
perfect  gas, 

(1)         v^  pi  =  v^Y  po,  or  =  pi  /  po. 


41]  RATIO  OF  THE  TWO  SPECIFIC  HEATS  45 

The  change,   from   I   to   III   was  isothermal;   hence,   by 
Boyle's   law, 

(  v2  }  7         (  pl  )  7 
(2)  ViPi=v>p2,or       j^J       -    j-j 

Hence  (pi  /  p^  =  (pi  /  p0)  ;    or,  taking  the  logarithm 
and  solving  for  7 


log  pi  -  log  pz 

The  desired  ratio  may  be  obtained,  experimentally, 
therefore,  by  observing  the  values  of  the  three  pressures. 

The  apparatus  consists  of  a  large  carboy  with  the  open- 
ing ground  flat,  and  covered  with  a  piece  of  ground  glass. 
The  enclosed  space  may  at  pleasure  be  opened  to,  or  shut 
off  from,  the  atmosphere.  The  pressure  of  the  enclosed 
air  is  measured  by  an  oil  manometer,  whilst  air  can  be 
forced  into  or  withdrawn  through  another  inlet.  To 
thoroughly  dry  the  enclosed  air,  some  strong  sulphuric 
acid  is  poured  into  the  bottom  of  the  carboy. 

(a)  Close  the  carboy,  and  with  a  bicycle  pump  intro- 
duce enough  air  in  the  carboy  to  give  a  reasonably  large 
difference  of  pressure,   as  indicated  by  the  manometer. 
Shut  off  connection  between  the  carboy  and  the  pump, 
and  wait  a  few  minutes  until  the  temperature  of  the  en- 
closed air  is  the  same  as  that  of  the  room,  which  will  be 
when  the  manometer  shows  a  steady,  constant  pressure. 
Read  the  manometer  and  the  barometer.     To  get  the 
value  of  the  pressure-difference  recorded  by  the  manom- 
eter, it  will  be  necessary  to  know  the  density  of  the  oil 
used.    This  is  posted  on  the  apparatus. 

(b)  Open  the  carboy  for  a  second,  by  removing  the 
glass  plate,  then  close  again.     Wait  some  time  until  the 


46  THE  HEAT  OF  FUSION  OF  TIN  [41-42 

temperature  of  the  enclosed  air  has  risen  again  to  that 
of  the  room,  as  indicated  by  a  steady,  constant  difference 
in  pressure;  then  read  the  manometer. 

(c)     Repeat  (a)  and  (b)  twice. 


(d)  Using  the  data  in  (a)  and  (6),  determine  from 
equation  (3)  the  value  of  y  for  air,  and  take  the  mean  of 
the  results. 

Obtain  from  the  Tables  the  value  of  7  and  compare 
with  the  result  just  found  by  experiment. 

Explain  why  the  specific  heat  of  a  gas  at  constant 
pressure  should  be  greater  than  its  specific  heat  at  con- 
stant volume. 


42.     THE  HEAT  OF  FUSION  OF  TIN. 

The  object  of  this  exercise  is  the  determination  of  the  heat 
of  fusion  of  tin  from  the  average  of  the  rate  of  cooling  just 
before  and  just  after  solidification. 

The  heat  of  fusion  of  any  substance  is  defined  as  the 
number  of  calories  required  to  convert  one  gram  of  the 
solid  at  the  melting  point  into  liquid  at  the  same  tem- 
perature. 

A  single  nickel-iron  thermocouple  is  connected  with  a 
sensitive  galvanometer  to  determine  the  temperature, 
the  galvanometer  deflections  being  converted  into  tem- 
peratures by  means  of  a  calibration  curve  based  upon  the 
deflection  of  the  galvanometer  for  four  standard  temper- 
atures: the  melting  point  of  zinc  419°C.,  the  melting 
point  of  tin  232°C.,  the  boiling  point  of  water  100°C.,  and 
0°C. 


42]  THE  HEAT  OF  FUSION  OF  TIN  47 

(a)  Connect  the  thermocouple  in  series  with  the  gal- 
vanometer. Put  one  of  the  junctures  in  a  beaker  of 
shaved  ice  and  the  other  in  a  small  graphite  crucible 
containing  a  small  quantity  of  pure  zinc.  After  the  zinc 
has  been  melted  allow  it  to  cool  off  by  radiation.  Arrange 
a  suitable  shunt  for  the  galvanometer,  or  insert  sufficient 
resistance  in  series  so  that  when  the  deflection  becomes 
constant,  as  it  will  during  the  time  of  solidification  of 
the  zinc,  this  scale  deflection  is  as  large  as  can  conven- 
iently be  obtained  on  the  galvanometer  scale.  After  this 
adjustment,  the  resistance  of  the  circuit  should  remain 
unchanged  throughout  the  rest  of  the  exercise. 

(6)  Determine  the  mass  of  the  tin  from  the  weight 
of  the  crucible  and  the  tin,  and  the  given  value  for  the 
weight  of  the  crucible.  Lift  the  juncture  from  the  zinc, 
taking  care  to  remove  whatever  particles  of  zinc  cling  to 
it.  Place  this  juncture  in  a  small  crucible  containing  at 
least  100  grams  of  pure  tin,  heating  it  up  until  the  gal- 
vanometer deflection  is  about  as  large  as  that  obtained 
with  the  zinc.  Put  out  the  burner  and  allow  the  crucible 
to  cool  by  radiation  when  placed  in  a  holder  with  a  three- 
point  asbestos  contact,  reading  the  scale  deflection  every 
30  seconds.  Take  five  or  six  readings  after  the  time  of 
constant  deflection,  which  indicated  the  temperature  of 
solidification  of  the  tin. 

(c)  Lift  the  juncture  from  the  tin,  place  it  in  a  small 
beaker  of  boiling  water  and  note  the  deflection.  Take 
a  final  reading  with  both  junctures  placed  in  shaved  ice. 


(d)  Make  a  plot  showing  the  relation  between  tem- 
perature and  the  galvanometer  deflections,  using  the 
constant  deflections  obtained  in  (a),  (6),  and  (c),  which 


48  THE  HEAT  OF  FUSION  OF  TIN  [42 

correspond  to  the  temperatures  419°,  232°,   100°,  0°  re- 
spectively.    Draw  a  smooth  curve  through  these  points. 

With  the  aid  of  the  above  plot  find  the  temperatures 
corresponding  to  each  scale  deflection  contained  in  (b). 
Make  a  plot  with  temperatures  as  ordinates  and  times  as 
abscissae,  using  the  data  in  (b)  for  the  cooling  of  the  tin. 

From  the  curve  find  the  rate  of  cooling  in  seconds  per 
degree  from  the  plot,  using  a  line  drawn  tangent  to  the 
curve  just  before  solidification,  and  a  second  value  from 
a  line  tangent  to  the  curve  just  after  solidification.  From 
the  heat  capacity  of  the  crucible  and  the  fluid  tin,  find 
the  rate  of  cooling  in  calories  per  second  just  before 
solidification.  And  using  the  heat  capacity  of  the  crucible 
and  that  of  solid  tin,  find  a  similar  value  for  the  rate  of 
cooling  just  after  solidification.  From  the  average  rate 
of  cooling  and  the  time  as  taken  from  the  plot,  find  the 
total  number  of  calories  given  out  during  solidification, 
and  calculate  the  heat  of  fusion  of  the  tin.  Specific  heat 
of  graphite  is  0.200,  of  porcelain  0.225,  of  fireclay  0.152, 
of  quartz  0.200,  of  fluid  tin  0.0637,  of  solid  tin  0.0588, 
calories  per  gram  respectively. 


PHYSICAL  TABLES 


49 


USEFUL  NUMERICAL  RELATIONS. 
Mensuration. 

Circle:  circumference  =  lirr;  area  =  TTr2. 
Sphere:  area  =  47Tr2;  volume  =  ^TTr3. 
Cylinder:  volume  =  TTr2/. 

Angle. 

1  radian  =  57°.2958  =  3437'.75. 
1  degree  =  0.017453  radian. 

Length. 

1  centimeter  (cm.)  =  0.3937  in.  1  inch  (in.)    =  2.540  cm. 

1  meter  (m.)  =  3.281  ft.  1  foot  (ft.)    =  0.3048  m. 

1  kilometer  (km.)    =  0.6214  mi.  1  mile  (mi.)  =  1.609  km. 

1  micron  (M)  =  0.001  mm.  1  mil  =  0.001  in. 


1  sq.  cm. 
1  sq.  m 


0.1550  sq.  in. 
10.674  sq.  ft. 


Area. 


Volume. 


1  cc.  =  0.06103  cu.  in. 

1  cu.  in.  =  35.317  cu.  ft. 

1  liter  (1000  cc.)  =  1.7608  pints. 


Mass. 


1  gram  (gm.) 
1  kilogram  (kg.) 


=  15.43  gr. 
=  2.2046  Ib. 


1  sq.  in.  =  6.451  sq.  cm. 
1  sq.  ft.  =  0.09290  sq.  m. 


1  cu.  in  =  16.386  cc. 

1  cu.  ft.  =  0.02832  cu.  m. 

1  quart  =  1.1359  liters. 


1  grain  (gr.)    =  0.06480  gm. 
1  pound  (Ib.)  =  0.45359  kg. 


Density* 

1  gm.  per  cc.    =  62.425  Ib.  per  cu.  ft. 
1  Ib.  per  cu.  ft.  =  0.01602  gm.  per  cc. 

Thermometric  Scales. 

C=5(F  — 32)/9  F  =  (9C/5)   4   32 

(C=  centigrade  temperature;  F  —  Fahrenheit  temperature) 


50  PHYSICAL  TABLES 


USEFUL  NUMERICAL  RELATIONS. 

Force. 

1  gram's  weight  (gm.  wt.)  =     980.6  dynes  (g0  =  980.6  cm./sec.1.) 
1  pound's  weight  (Ib.  wt.)    =  0.4448  megadynes  (g0  =  980.6.) 

(The  "gm.  wt.'  is  here  defined  as  the  force  of  gravity  acting  on 
a  gram  of  matter  at  sea-level  and  45°  north  latitude.  The  "Ib.  wt."  is 
similarly  defined.) 

Pressure  and  Stress 

1  cm.  of  mercury  at  0°C.  1  in.  of  mercury  at  0°C. 

=  13.596  gm.  wt.  per  sq.  cm.  =  34.533  gm.  wt.  per  sq.  cm. 

=  0.19338  Ib.  wt.  per  sq.  in.  =  0.49118  Ib.  wt.  per  sq.  in. 

Work  and  Energy. 

1  kilogram-meter  (kg.  m.)  =  7.233  ft.  Ib. 

1  foot-pound  (ft.  Ib.)  =  0.13826  kg.  m. 

1  joule  =  107  ergs. 

1  foot-pound  =  1.3557  X  107  ergs. (&=  980.6  cm./sec.1.) 

1  foot-pound  =  1.3557  joules  (g0  =  980.6.) 

1  joule  =  0.7376  ft.  Ib.  (g0  =  980.6.) 

Power  (or  Activity). 

1  horse-power  (H.  P.)  =  33000  ft.  Ib.  per  min. 

1  watt  =  1  joule  per  sec.  =  107  ergs  per  sec. 

1  horse-power  =  745.64  watts  (g0  =  980.6  cm./sec.1) 

1  watt  =  44.28  ft.  Ib.  per  min  (g0  =  980.6) 

Mechanical  Equivalent. 

1  gm.-calorie  =  4.187  X  107  ergs. 

=  0.4269  kg.  m.  (g0  =  980.6  cm./sec.2.) 
=  3.088  ft.  Ib.  (g0  =  980.6.) 


PHYSICAL  TABLES 


51 


DENSITY  OF  DRY  AIR. 

(Values  are  given  in  gms.  per  cc.) 


Temp. 

Barometric  Pressure 

(Centimeters  of  Mercury) 

C. 

72 

73 

74 

75 

76 

77 

0° 

.001225!  .001242 

.001259 

.001276 

.001293 

.001310 

1 

220 

237 

254 

271 

288 

305 

2 

.  216 

233 

250 

267 

283 

300 

3 

212 

228 

245 

262 

279 

296 

4 

207 

224 

241 

257 

274 

290 

5° 

.001203 

.001219 

.001236 

.001253 

.001270 

.001286 

6 

198 

215 

232 

248 

265 

282 

7 

194 

211 

227 

244 

260 

277 

8 

190 

206 

223 

239 

256 

272 

9 

186 

202 

219 

235 

251 

268 

10° 

.001181 

.001198 

.001214 

.001231 

.001247 

.001263 

11 

177 

194 

210 

226 

243 

259 

12 

173 

189 

206 

222 

238 

255 

13 

169 

185 

202 

218 

234 

250 

14 

165 

181 

197 

214 

230 

246 

15° 

.001161 

.001177 

.001193 

.001209 

.001225 

.001242 

16 

157 

173 

189 

205 

221 

237 

17 

153 

169 

185 

201 

217 

233 

18 

149 

165 

181 

197 

213 

229 

19 

145 

161 

177 

193 

209 

224 

20° 

.001141 

.001157 

.001173 

.001189 

.001204 

.001220 

21 

137 

153 

169 

185 

200 

216 

22 

133 

149 

165 

181 

196 

212 

23 

130 

145 

161 

177 

192 

208 

24 

126 

141 

157 

-   173 

188 

204 

25° 

.001122 

.001138 

.001153 

.001169 

.001184 

.001200 

26 

118 

134 

149 

165 

180 

196 

27 

114 

130 

145 

161 

176 

192 

28 

110 

126 

142 

157 

172 

188 

29 

107 

122 

138     153 

169 

184 

30° 

.001103 

.001119 

.0011341  .001149 

.  01165 

.001180 

Corrections 

for  Moisture  in  the  Atmosphere 

Dew-point  Subtract!  Dew-point 

Subtract  Dew-point 

Subtract 

—10°   1  .000001 

+  2° 

.000003     -i-  14° 

.000007 

—  8         2 

+  4 

4  '    +16 

8 

—  6         2 

+  6 

4     +18 

9 

—  4         2 

+  8 

'  5     +20 

.000010 

—  2         3 

+  10 

6     +24 

13 

0         3 

+  12 

6     +28 

16 

52 


PHYSICAL  TABLES 


DENSITIES    AND    THERMAL    PROPERTIES    OF    GASES. 


(The  densities  are  given  at  0°C.  and  76  cm.  pressure,  and  the 
specific  heats  at  ordinary  temperatures.  The  coefficients  of  cubical 
expansion  (at  constant  pressure)  of  the  gases  listed  below  are  not 
given  in  this  Table;  they  are  about  the  same  for  all  the  permanent 
gases,  being  approximately  1/273  or  0.003663,  if  referred  in  each 
case  to  the  volume  of  the  gas  at  0°C.  The  specific  heats  at  constant 
pressure  and  at  constant  volume  are  represented  by  the  symbols 
Sp.  and  Sv) . 


Gas  or  Vapor 

Formula 

Density 
(gms.  per  cc.) 

Molecular 
Weight 

Sp  ~j~  Sv 

SP 

(cals. 
per  gm.) 

A  it- 

f)  f)()1  OQQ 

4.1 

0907 

Ammonia 
Carbon  dioxide 
Carbon  monoxide 
Chlorine 
Hydrochloric  acid 
Hydrogen 
Hydrogen  sulphide 
Nitrogen,  pure 

NH3 
C02 
CO 
C12 
HC1 
H2 
H2S 
N2 

0.000770 
0.001974 
0.001234 
0.003133 
0.001616 
0.0000896 
0.001476 
0.001254 

ft  OO1  9^7 

17.06 
44.00 
28.00 
70.90 
36.46 
2.016 
34.08 
28.08 

.33 
.29 
.40 
.32 
.40 
.41 
1.34 
1.41 

.530 
.203 
.243 
.124 
.194 
3.410 
.245 
.244 

Oxygen 
Steam  (100°C.) 
Sulphur  dioxide 

02 
H2O 
S02 

0.001430 
0.000581 
0.002785 

32.00 
18.02 
64.06 

1.41 
1.28 
1.26 

.218 
.421 
.154 

DENSITY  AND  SPECIFIC  VOLUME  OF  WATER. 


Temp. 
C. 

Density 
(gms.  per  cc.) 

Specific 
Volume 

(cc.  per  gm.) 

Temp. 
C. 

Density 
(gms.  per  cc.) 

Specific 
Volume 

(cc.  per  gm.) 

0 

0.999868 

1.000132 

20° 

0.99823 

1.00177 

1 

927 

073 

25 

777 

294 

2 

968 

032 

30 

567 

435 

3 

992 

008 

35 

406 

598 

3.98 

1.000000 

000 

40 

224 

782 

5 

.999992 

008 

50 

.98807 

1.01207 

6 

968 

032 

60 

324 

705 

7 

929  I             071 

70 

.97781 

1.02270 

8 

876  !            -124 

80 

183 

902 

9 

808 

192 

90 

.96534 

1.03590 

10 

727 

273 

100             .95838 

1.04343 

15 

126 

874 

102 

693 

501 

PHYSICAL  TABLES 


53 


DENSITIES   AND   THERMAL   PROPERTIES   OF   LIQUIDS. 

(The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  liquids  listed.  The  student  should  not  expect  the  properties 
of  the  average  laboratory  specimen  to  correspond  exactly  in  value 
with  them.  With  a  few  exceptions  the  densities  are  given  for  ordi- 
nary atmospheric  temperature  and  pressure.  The  specific  heats  and 
coefficients  of  expansion  are  in  most  cases  the  average  values  be- 
tween 0°  and  100°C.  The  boiling  points  are  given  for  atmospheric 
pressure,  and  the  heats  of  vaporization  are  given  at  these  boiling 
points.) 


•+-* 

0 

a  *3  9 

<u    o  .2 

bo 

n 

"ej      rt 

*o  'J3    2 

3  o 

O    rt 

Liquid 

a 

a> 
Q 

C& 

1^   | 

!'! 

o   o 

(calories 

> 

per  gm. 

(per  degree 

(degrees 

(  calories 

(gms.  per  cc.) 

per  deg.) 

C.) 

C.) 

per  gm.) 

Alcohol  (ethyl) 

0.794 

.58 

.00111 

78 

205* 

Alcohol  (methyl) 

.796 

.60 

.00143 

66 

262t 

Benzene 

.880 

.42 

.00123 

80 

93.2 

Carbon  bisulphide 

1.29 

.24 

.00120 

46.6 

84 

Cotton  seed  oil 

.925 

.47 

.00077 

Ether 

.74  (0°C) 

.55 

.00162 

35 

90 

Glycerine 

1.26 

.576 

.000534 

Hydrochloric  acid 

1.27 

.75 

.000455 

110 

Mercury 
Olive  oil 

13.596  (0°) 
.918 

.033 
.47 

.0001815 
.000721 

357 

67 

Citric  acid 

1.56 

.66 

.00125 

86 

115 

Sea-water 

1.025 

.938 

Sulphuric  acid 

1.85 

.33 

.00056 

338 

122 

Turpentine 

.873 

.47 

.00105 

159 

70 

*  The  heat  of  vaporization  of  ethyl  alcohol  at  0°C.  is  236.5. 
t  The  heat  of  vaporization  of  methyl  alcohol  at  0°C.  is  289.2. 


54 


PHYSICAL  TABLES 


DENSITIES    AND    THERMAL    PROPERTIES    OF    SOLIDS. 

(The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  substances  listed.  The  student  should  not  expect  the  prop- 
erties of  the  average  laboratory  specimen  to  correspond  exactly 
in  value  with  them.  As  a  rule  the  densities  are  given  for  ordi- 
nary atmospheric  temperature.  The  specific  heats  and  coefficients 
of  expansion  are  in  most  cases  the  average  values  between  0°  and 
100°C.  The  melting  points  and  heats  of  fusion  are  given  for 
atmospheric  pressure.) 

The  coefficient  of  cubical  expansion  of  solids  is  approximately 
three  times  the  linear  coefficient. 


* 

o 

111 

<L>  bo 
W)  g  -*-• 

*°   0 

1 

1? 

o.S  S 

g|| 

rt  *^ 

Solid. 

1 

aW 

CO 

1*1 

|3* 

5  3 

cals.  per 

degrees 

cals.  per 

gms.  per  cc. 

gm. 

per  degree  C. 

C. 

gm. 

Acetamide 

1.56 

82 

Aluminum 

2.70 

0.219 

.0000231 

658 

Brass,  cast 

8.44 

.092      1.  0000188 

"       drawn 

8.70 

.092       .0000193 

Copper 

8.92 

.094 

.0000172 

1090 

43.0 

German-silver 

8.62 

.0946    L000018 

860 

Glass,  common  tube 

2.46 

.186      i.  0000086 

"       flint 

3.9 

.117       .0000079 

Gold 

19.3 

.0316     .0000144 

1065 

Hyposul.  of  soda 

1.73 

.445 

48 

Ice 

.918 

.502 

.000051 

0 

80. 

Iron,  cast 

7.4 

.113 

.0000106 

1100 

23-33 

"     wrought 

7.8 

.115 

.000012 

1600 

Lead 

11.3 

.0315     .000029 

326 

5.4 

Mercury 

13.596 

.0319 

—39 

2.8 

Nickel 

8.90 

.109 

.0000128 

1480 

4.6 

Paraffin,  wax 

.90 

.560 

.000008-23 

52 

35.1 

liquid 

.710 

Platinum 

21.50 

.0324    .0000090 

1760 

27.2 

Rubber,  hard 

1.22 

.331 

.000064 

Silver 

10.53 

.056 

.0000193 

960 

21.1 

Sodium  chloride 

2.17 

.214 

.000040 

800 

Steel 

7.8 

.118 

.000011 

1375 

Tin 

7.29 

.0588 

.0000214 

232 

14. 

Wood's  alloy,  solid 

9.78 

.0352 

75.5 

8.40 

"          "     ,  liquid 

.0426 

PHYSICAL  TABLES 


55 


SURFACE  TENSION  OF  PURE  WATER  IN  CONTACT 
WITH  AIR. 

(The  value  of  the  surface  tension  of  a  liquid  is  dependent  only 
upon  the  character  and  temperature  of  the  liquid  and  upon  the 
nature  of  the  gas  above  the  surface  of  the  liquid.  It  is  independent  of 
the  curvature  of  the  surface  film  and  of  the  material  of  the  con- 
taining vessel.) 


Temp. 
C. 

Tension 
(dynes  pr.  cm) 

Temp. 
C. 

Tension 
(dynes  pr.  cm) 

Temp. 
C. 

Tension 
(dynes  pr.cm) 

0° 
5 
10 
15 
20 
25 

75.5 
74.8 
74.0 
73.3 
72.5 
71.8 

30° 
35 
40 
45 
50 
55 

71.0 
70.3 
69.5 
68.6 
67.8 
66.9 

60° 
65 
70 
80 
100 

crit.  temp 

66.0 
65.1 
64.2 
62.3 
56.0 
0.0 

SURFACE   TENSIONS   OF   SOME   LIQUIDS   IN   CONTACT 
WITH  AIR. 

(The  angle  of  contact  between  pure  water  and  clean  glass  vessels 
of  all  sizes  is  0°;  the  angle  of  contact  between  pure  water  and  clean 
steel  or  silver  is  about  90°;  the  angle  of  contact  between  mercury 
and  glass  is  about  132°.  See  the  note  to  Table  VIII.) 


Dynes 
per  cm. 

Dynes 
per  cm. 

Alcohol  (ethyl)      at  20° 
Alcohol  (methyl)  at  20° 
Benzene                   at  15° 
Glycerine                at  18° 

22-24 
22-24 
28-30 
63-65 

Mercury         at  20° 
Olive  oil          at  20° 
Petroleum      at  20° 
Water  (pure)  at  20° 

470-500 
32-36 
24-  26 

72-  74 

VISCOSITY  OF  WATER. 


Temp.  Coeff  .  of  Vise. 

C.      (C.G.S.Units) 

; 

Temp. 
C. 

Coeff.  of  Vise. 

(C.G.S.Units) 

Temp. 
C. 

Coeff.  of  Vise. 
(C.G.S.Units) 

0°            0.0178 
5                .0151 
10                .0131 
15                .0113 
20                .0100 

25° 
30 
35 
40 
50 

0.0089 
.0080 
.0072 
.0066 
.005- 

60° 
70 
80 
90 
100 

0.0047 
.0041 
.0036 
.0032 
.0028 

56 


PHYSICAL  TABLES 


(a)  BOILING  POINT  OF  WATER  AT  DIFFERENT  BARO- 

METRIC PRESSURES. 

(b)  VAPOR-PRESSURE  OF  SATURATED  WATER-VAPOR. 

(This  table  may  be  used  either  (a)  to  find  the  boiling  point  /  of 
water  under  the  barometric  pressure  P,  or  (b)  to  find  the  vapor- 
pressure  P  of  water-vapor  saturated  at  the  temperature  t,  the  dew- 
point.) 


t° 
c. 

P 
cm. 

D 
gm./cc 

t° 
C. 

P 
cm. 

D 
gm./cc. 

t° 
C. 

P 
cm. 

D 

gm./cc. 

-10°j  .22 

2.3xlO-6 

30°  :  3.15 

30.1x10- 

88.5  49.62 

-  9   .23 

2.5x  " 

35   4.18  39.3x' 

89 

50.58 

-  8 

.25 

2.7x" 

40   5.49 

50.9x  ' 

89.5  51.55 

-  7 

.27 

2.9x  " 

45   7.14  65.3x  ' 

90 

52.54 

428.4xlO-6 

-  6 

.29 

3.2x  " 

50   9.20 

83.0x  ' 

90.5  53.55 

-  5 

.32 

3.4x  " 

55  111.75 

104.6x  ' 

91 

54.57 

-  4 

.34 

3.7x  " 

60 

14.88 

130.7x  « 

91.5 

55.61 

-  3 

.37 

4.0x  " 

65 

18.70 

162.1x' 

92 

56.67 

-  2 

.39 

4.2x  " 

70 

23.31 

199.5x  ' 

92.5 

1  57.74 

-  1 

.42 

4.5x  " 

71 

24.36 

93 

i  58.83 

0 

.46 

4.9x  " 

72 

25.43 

93.5 

59.96 

1 

.49 

5.2x  " 

73 

26.54 

94 

61.06 

2 

.53 

5.6x  " 

74 

27.69  i 

94.5 

62.20 

3 

.57 

6.0x  ' 

75 

28.88 

243.7" 

95 

63.36 

511.1" 

4 

.61 

6.4x  ' 

75.5 

29.49 

95.5 

64.54 

5 

.65 

6.8x  ' 

76 

30.11 

96 

65.74 

6 

.70 

7.3x  ' 

76.5 

30.74 

96.5 

66.95 

7 

.75 

7.7x  ' 

77 

31.38 

97 

68.18 

8 

.80 

8.2x  ' 

77.5 

32.04 

97.5 

69.42 

9 

.85 

8.7x  ' 

78 

32.71 

98 

70.71 

10   .91 

9.3x  " 

78.5 

33.38 

98.2!  71.23 

11 

.98 

lO.Ox  " 

79 

34.07 

98.4 

71.74 

12 

1.04 

10.6x  " 

79.5 

34.77 

98.6 

72.26 

13 

1.11 

11.2x" 

80 

35.49 

295.9  " 

98.8 

72.79 

14 

1.19 

12.0x  " 

80.5 

36.21 

99 

73.32 

15 

1.27 

12.8x  " 

81 

36.95 

99.2 

73.85 

16 

1.35 

13.5x" 

81.5 

37.70 

99.4 

74.38 

17 

1.44 

14.4x  " 

82 

38.46  1 

99.6 

74.92 

18 

1.53 

15.2x  " 

82.5 

39.24  ! 

99.8 

75.47 

19 

1.63 

16.2x  " 

83 

40.03 

100 

76.00 

606.2  " 

20 

1.74 

17.2x  " 

83.5 

40.83 

100.2 

76.55 

21 

1.85 

18.2x  " 

84 

41.65 

100.4 

77.10 

22 

1  5o 

19.3x  " 

84.5 

42.47 

00.6 

77.65 

23 

2.09  20.4x  " 

85 

43.32 

357.1  " 

00.8 

78.21 

24  2.2221.6x" 

85.5 

44.17 

01 

78.77 

25  2.35  22.9x  " 

86 

45.05 

02 

81.60 

26  2.50  24.2x  " 

86.5 

45.93 

03 

84.53 

27  !  2.65 

25.6x  " 

87 

46.83 

05 

90.64 

715.4  " 

28  2.81  27.0x  " 

87.5  47.74 

07 

97.11 

29  2.97  28.5x  " 

88  148.68 

10 

107.54 

840.1  " 

PHYSICAL  TABLES 


57 


THE  WET-  AND  DRY-  BULB  HYGROMETER.  DEW-POINT. 

(This  Table  gives  the  vapor-pressure,  in  mercurial  centimeters,  of 
the  water-vapor  in  the  atmosphere  corresponding  to  the  dry-bulb 
reading  /°C.  (first  column)  and  the  difference  (first  row)  between 
the  dry-bulb  and  wet-bulb  readings  of  the  hygrometer.  Having 
obtained  from  this  Table  the  value  of  the  vapor-pressure  for  a  given 
case,  the  dew-point  can  be  found  by  consulting  the  table.  The  data 
given  below  are  calculated  for  a  barometric  pressure  equal  to  76  cm.) 


*°c 

Difference  between  Dry-bulb  and  Wet-bulb  Readings. 

0°       1°        2°       3°       4°       5°       6°        7°        8°       9°      10° 

cm. 

cm.     cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

i 

10° 

.92 

.81 

.70 

.60 

.50 

.40      .31 

.22 

.13 

11 

.98 

.87 

.76 

.65 

.55 

.45      .35 

.26 

.17 

12 

.105 

.93 

.82 

.71 

.60 

.50      .40 

.30 

.21 

.12 

.03 

13 

1.12 

.00 

.89 

.76 

.65 

.55 

.45 

.35 

.25 

.16 

.07 

14 

1.19 

.07 

.94 

.83 

.71 

.61 

.50 

.40 

.30 

.20 

.11 

15 

1.27 

.14 

1.01 

.90 

.78 

.66 

-.55 

.45 

.34 

.25 

.15 

16 

1.35 

.22 

1.09 

.97 

.84 

.73 

.60 

.50 

.40 

.30 

.19 

17 

1.44 

.30 

1.17 

1.04 

.91 

.80      .67 

.56 

.45 

.35 

.24 

18 

1.54 

1.39 

1.25 

1.12 

.99 

.86      .74 

.63 

.51 

.40 

.30 

19 

1.63 

1.49 

1.34 

1.20 

1.07 

.94 

.81 

.69 

.57 

.46 

.35 

20 

1.74 

1.59 

.43    1.29 

1.15 

1.02 

.88 

.76 

.64 

.52 

.41 

21 

1.85 

1.69 

.53    1.38 

1.24 

1.10      .96 

.84 

.71 

.59 

.47 

22 

1.97 

1.80      .64 

1.48 

1.33 

1.19 

.05 

.91 

.78 

.66 

.54 

23 

2.09 

1.92 

.75 

1.59 

1.43 

1.28 

.13 

.00 

.86 

.73 

.61 

24 

2.22 

2.04 

.86 

1.70 

1.53 

1.38 

.231     .09 

.94 

.81 

.68 

25 

2.35 

2.17 

.99 

1.81 

1.64 

1.48 

.33|     .18 

1.03 

.90 

.76 

26 

2.50 

2.31 

2.1l{  1.94 

1.76 

1.59 

.43 

.28 

1.13 

.98      .84 

27 

2.65 

2.45 

2.25    2.07 

1.88 

1.71 

1.54 

.38 

1.23 

1.08 

.93 

28 

2.81 

2.60 

2.40;  2.20    2.01 

1.83 

1.66 

.49 

1.33 

1.18 

1.02 

29 

2.98 

2.76 

2.55    2.35 

2.15 

1.96 

1.78 

.61 

1.44 

1.28 

1.12 

30 

3.15 

2.931  2.71    2.50    2.29:  2.10    L9l]  1.73 

1.55    1.39 

1.23 

Miscellaneous. 

(1.)      Heat  of  Neutralization. 

Any  strong  acid  with  any  strong  alkali  evolves  (+)  about 

761-  calories  for  every  gm.  of  water  formed. 
(2.)      Heat  of  Solution  in  water. 

For  Calcium  oxide  (CaO),  -f-  327  cals.  per  gm. 

"    Sodium  chloride  (NaCl),  -     21     "       "     " 

hydroxide  (NaOH),  +  248     "       "     " 

hyposulphite  (Na2S2O3-f  5H2O),  -    44.8  "       "     " 

(3.)     Fuel  value  of  illuminating  gas  is  5500  to  6500  calories  per  liter, 
its  density  is  .00058  gm.  per  cc.  at  0°C  and  76  cm.  pressure. 

Fuel  value   of  ethyl  alcohol   is  7400,  of  methyl   alcohol    5700, 
calories  per  gram. 


58 


NATURAL  SINES. 


0' 

6 

12' 

18 

24' 

30' 

36 

42' 

48' 

54' 

123 

4      5 

0° 

~T~ 
2 
3 

oooo 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12      15 

0175 
0349 
0523 

0192 
0366 
0541 

0209 
0384 
0558 

0227 
0401 
0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
0663 

0332 
0506 
0680 

369 
369 
369 

12      15 
12      15 

12     15 

4 
5 
6 

0698 
0872 
1045 

0715 
0889 
1063 

0732 
0906 
1080 

0750 
0924 
1097 

0767 
0941 
i"5 

0785 
0958 
1132 

0802 
0976 
"49 

0819 

0993 
II6.7 

0837 
ion 
1184 

0854 
1028 

1  201 

369 
369 
369 

12      15 
12      14 
12      14 

7 
8 
9 

1219 

1392 
1564 

1236 
1409 
1582 

1253 
1426 

'599 

1271 
1444 
1616 

1288 
1461 
1633 

1305 
1478 
1650 

*323 
1495 
1668 

1340 
1513 
1685 

1357 
1530 
1702 

1374 
1547 
1719 

369 
369 
369 

12      14 
12      14 
12      14 

10 

1736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

369 

12      14 

11 
12 
13 

1908 
2079 

2250 

1925 
2096 
2267 

1942 
2113 

2284 

1959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

201  1 

2181 

2351 

2028 
2198 
2368 

2045 
2215 
2385 

2062 
2232 
2402 

369 
369 
368 

II       14 
II       14 
II       14 

14 
15 
16 

24*19 

2588 
2756 

2436 
2605 
2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 
2857 

2538 
2706 
2874 

2554 
2723 
2890 

2571 
2740 
2907 

368 
368 
368 

II       14 
II       14 
II       14 

17 
18 
19 

2924 
3090 
3256 

2940 
3107 
3272 

2957 
3123 
3289 

2974 
3140 

3305 

2990 
3156 
3322 

3007 
3173 
3338 

3024 
3190 

3355 

3040 
3206 
3371 

3057 
3223 

3387 

3074 
3239 
3404 

368 
368 
3  5  8 

II       14 
II       14 
II       14 

20 

3420 

3437 

3453 

3469 

3486 

3502 

35i8 

3535 

3551 

3567 

3  5  8 

II       14 

21 
22 
23 

3584 
3746 

39°7 

3600 
3762 
3923 
4083 
4242 
4399 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 
397i 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

3714 
3875 
4035 

3730 
3891 
4051 

3  5  8 
3  5  8 
3  5  8 

II       14 
II       14 
II       14 

24 
25 
26 

4067 
4226 
4384 

4099 
4258 
4415 

4H5 
4274 

4431 

*i3i 
4289 

H4& 

4M7 
4305 
4462 

4163 
4321 
4478 

4179 
4337 
4493 

4195 
4352 
4509 

4210 
4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II       13 
II       13 

10    13 

27 
28 

29 

4540 
4695 

4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
4741 
4894 

4602 
4756 
t909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
4970 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10    13 
10    13 
10    13 

30 

5000 
5150 
5299 
5446 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

3  5  8 

10    13 

31 
32 
33 

5165 
53M 
5461 

5180 
5329 
5476 

5195 
5344 
5490 

5210 

5358 
5505 

5225 
5373 
5519 

5240 

5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 

5577 

2  5  7 
257 
257 

10      12 
10      12 
10      12 

34 
35 
36 

5592 
5736 

5878 

5606 
5750 
5892 

5621 
5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

5707 
5850 
599° 

5721 
5864 
6004 

2  5.  7 
257 
2  5  7 

10      12 
10      12 

9     12 

37 
38 
39 

6018 

6i57 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 

6347 

6088 
6225 
6361 

6101 
6239 
6374 

6115 
6252 
6388 

6129  6143 
6266  6280 
6401  6414 

257 
257 
247 

9    12 

9      II 

9     'I 

40 

6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

247 

9     ii 

41 

42 
43 

6561 
6691 
6820 

6574 
6704 

6833 

6587 
6717 
6845 

6600 
6730 

6858 

6613 
6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

6652 
6782 
6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
246 
246 

9     II 
9    ii 
8     ii 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

246 

8    10 

NATURAL  SINES. 


59 


0' 

6' 

12 

18 

24 

30 

36 

42 

48 

54 

123 

4  5 

45° 

~46~ 
47 
48 

7071 

7083 

7096 

7108 

7120 

7133 

7M5 

7157 

7169 

7181 

2  4  6 

8  10 

7193 
7314 
743i 

7206 
7325 
7443 

7218 
7337 

7455 

7230 
7349 
7466 

7242 
7361 

7478 

7254 
7373 
7490 

7266 

7385 
750i 

7278 
7396 
7513 

7290 
7408 
7524 

7302 
7420 
7536 

246 
246 
246 

8  10 
8  10 
8  10 

49 
50 
51 
~52~ 
53 
54 

7547 
7660 

7771 

7558 
7672 
7782 

7570 
7683 
7793 
7902 
8007 
8m 

758i 
7694 
7804 

7593 
7705 
T^IS 

7923 
8028 
8131 

7604 
7716 
7826 

7615 

7727 
7837 

7627 
7738 
7848 

7638 
7749 
7«59 

7649 
7760 
7869 

246 

246 
245 

8  9 
7  9 
7  9 

7880 
7986 
8090 

7891 

7997 
8100 

7912 
8018 
8121 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

245 
2  3  5 
2  3  5 

7  9 
7  9 
7  8 

55 

8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

2  3  5 

7  8 

56 
57 
58 

829Q 

8387 
8480 

8572 
8660 

8746 
8829 

8910 

8988 

8300 
8396 
8490 

8581 
8669 
8755 
8838 
8918 
8996 

8310 
8406 
8499 
8590 
8678 
8763 

8320 
8415 
8508 

8329 
8425 

8517 

8339 
8434 
8526 

8348 
8443 
8536 

8358 
8453 

8545 

8368 
8462 
8554 

8377 
8471 

8563 

235 
2  3  5 
2  3  5 

6  8 
6  8 
6  8 

59 
60 
61 

8599 
8686 
8771 

8607 
8695 
8780 

8616 
8704 

8788 

8625 
8712 
8796 

8634 
8721 
8805 

8643 
8729 
8813 

8652 
8738 
8821 

i  3  4 
i  3  4 
i  3  4 

6  7 
6  7 
6  7 

62 

63 
64 

8846 
8926 
9QQ3 
9078 

8854 
8934 
9011 

8862 
8942 
9018 

8870 

8949 
9026 

8878 
8957 
9033 

8886 
8965 
9041 

8894 
8973 
9048 

8902 
8980 
9056 

i  3  4 

i  3  4 
i  3  4 

5  7 
5  6 
5  6 

65 

9063 

9070 

9143 
9212 
9278 

9085 

9092 
9164 
9232 
9298 

9100 

9i7i 
9239 
9304 

9107 

9114 

9121 

9128 

I  2  4 

5  6 

66 
67 
68 

9135 

9205 

9272 

9'50 
9219 
9285 

9157 
9225 
9291 

9178 
9245 
93U 

9184 
9252 
9317 

9191 
9259 
9323 

919^ 
9265 
9330 

I  2  3 
I  2  3 
I  2  3 

5  6 
4  6 
4  5 

69 
70 
71 
72 
73 
74 

9336 

9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
9415 
9472 

9361 
9421 

9478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 

939  i 
9449 
9505 

I  2  3 

I  2  3 

i  2  3 

4  5 
4  5 
4  5 

9500 

95U 
9563 
9613 

>5i6 

9568 
9617 

952i 
9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9548 
9598 
9646 

9553 
9603 
9650 

9558 
9608 
9655 

I  2  3 

I  2  2 
122 

4  4 
3  4 
3  4 

75 
76 
77 
78 

9659 

0664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

112 

3  4 

9703 
9744 
9781 

9707 
9748 
9785 

9711 

9751 
9789 

9715 
9755 
9792 

9720 
9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9736 
9774 
9810 

9740 
9778 
9813 

112 
112 
112 

3  3 
3  3 

2  3 

79 
80 
81 

9816 
9848 
9877 

9820 

9851 
9880 

9823 

9854 
9882 

9826 
9857 
9885 

9829 
)86o 
)888 

)833 
9863 
9890 
9914 
9936 
9954 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I  2 
O  I 

o 

2  3 
2  2 
2  2 

82 
83 
84 

9903 
9925 
9945 

9905 
9928 

9947 

9907 
9930 
9949 

9910 
9932 
995i 

9912 
9934 
9952 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

O 
O 
O 

2  2 
I  2 
I  I 

85 

9962 

9963 

9965 

9966 

9968 

9969 

997i 

9972 

9973 

9974 

0  0 

I  I 

86 
87 
88 

9976 
9986 
9994 

9977 
9987 
9995 

9978 
9988 
9995 

9979 
9989 
9996 

9980 
9990 
9996 

9981 
9990 
9997 

9982 
9991 
9997 

9983 
9992 
9997 

9984 

9993 
9998 

9985 
9993 
9998 

0     0 
O     O      O 
O     O      O 

I  I 
I  I 
0  0 

89 

9998 

9999 

9999 

9999 

9999 

1.000 
nearly 

1.  000 
nearly 

1.000 

nearlv 

1.  000 
nearly 

1.  000 
nearly 

o  o  o 

O  O 

60 


NATURAL  TANGENTS. 


0' 

6 

12' 

18 

24' 

30' 

36 

42 

48' 

54' 

123 

4      5 

0° 

.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12         14- 

1 

2 
3 

.0175 

.0349 
.0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
0577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 
0454 
0629 

0297 
0472 
0647 

0314 
0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12         15 

12         15 

12          I5 

4 
5 
6 

.0699 

.0875 
.1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 

"39 

0805 
0981 
"57 

0822 
0998 
H75 

0840 
1016 
1192 

0857 
1033 

1210 

369 
369 
369 

12          15 
12          15 
12          I5 

7 
8 
9 

.1228 
.1405 
.1584 

1246 
1423 
1602 

1263 
1441 
1620 

1281 

1459 
1638 

1299 
1477 
1655 

1317 
1495 
1673 

1334 
1512 
1691 

f352 
1530 
1709 

1370 
1548 
1727 

1388 
1566 
1745 

369 

3  6  9 
369 

12         15 
12          15 
12          I5 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

369 

12          15 

11 
12 
13 

.1944 
.2126 

.2309 

1962 
2144 
2327 

1980 
2162 
2345 

1998 
2180 
2364 

2016 
2199 

2382 

2035 
2217 
2401 

2053 

2235 
2419 

2071 
2254 
2438 

2089 
2272 
2456 

2107 
2290 
2475 
266l 
2849 
3038 

369 
3  6  9 
369 

12          15 
12          15 
12          15 

14 
15 
16 

.2493 
.2679 
.2867 

2512 
2698 
2886 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 
3172 
3365 
356i 

2623 
28ll 
3000 

2642 
2830 
3019 

369 
369 
369 

12          l6 

13          '<< 

13       1  6 

17 
18 
19 

.3057 
•3249 

•3443 

3076 
3269 
3463 

3096 

3288 
3482 

3"5 
3307 
3502 

3134 
3327 
3522 

3153 
3346 
3541 

3I91 

3385 
3581 

3211 
3404 
3600 

3230 
3424 
3620 

3  6  10 
3  6  10 
3  6  10 

13       16 
13       »6 
13       '7 

20 

•3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3  7  '0 

13       '7 

21 
22 
23 

.3839 
.4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

4307 

3919 
4122 

4327 

3939 
4142 

4348 

3959 
4163 
4369 

3978 
4183 
4390 

4000 
4204 
4411 

4O2O 
4224 
4431 

3  7  >o 
3  7  10 
3'  7  Jo 

'3       »7 
»4       i7 
14       17 

24 
25 
26 

•4452 
.4663 
.4877 

4473 
4684 
4899 

4494 
4706 
4921 

4515 
4727 
4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 
4791 
5008 

4599 
4813 
5029 

4621 
4834 
5051 

4642 
4856 
5073 

4  7  ii 
4  7  ii 

,4      18 

15       18 

27 
28 
29 

•5095 
•5317 

•5543 

5U7 
5340 
5566 

5139 
5362 
5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

5250 
5475 
5704 

5272 
5498 
5727 

5295 
5520 
5750 

4  7  I' 

4  8  ii 
4  8  12 

15       18 
15       '9 
'5       19 

30 

•5774 

5797 

5820 

5844 

5867 

5890 

5914 

5938 

596i 

5985 

4  8  12 

l6         20 

31 
32 
33 

.6009 
.6249 
.6494 

6032 
6273 
6519 

6056 
6297 
6544 

6080 
6322 
6569 

6lO4 
6346 
6594 

6128 
6371 
6619 

6152 
6395 
6644 
6899 
7159 
7427 

6176 
6420 
6669 
6^24 
7186 
7454 

6200 
6445 
6694 

6224 
6469 
6720 

4  8  12 
4  8  12 
4  §  13 

l6         2O 

16      20 

17         21 

34 
35 
36 

6745 
.7002 
.7265 

6771 
7028 
7292 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 

7373 

6873 
7133 
7400 

6950 
7212 
7481 

6976 

7239 
7508 

4  9  13 
4  9  '3 
5  9  '4 

17         21 

l8          22 
l8         23 

37 
38 
39 

•7536 
•7813 
.8098 

7563 
7841 
8127 

7590 
7869 
8156 

7618 
7898 
8185 

7646 
7926 

8214 

7673 
7954 
8243 

7701 
7983 
8273 

7729 
8012 
8302 

7757 
8040 
8332 

7785 
8069 
836! 

5  9  '4 
5  °  M 
5  o  15 

1  8      23 
19     24 

20         =4 

40 

.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

5  o  15 

20          25 

41 
42 
43 

.8693 
.9004 
•9325 

8724 
9036 
9358 

8754 
9067 

9391 

8785 
9099 
9424 

8816 
9131 
9457 

8847 
9163 
9490 

8878 
9195 
9523 

8910 
9228 
9556 

8941 
9260 
9590 

8972 
9293 
9623 

9965 

5016 
5  i  16 
6  .  17 

21          26 

zi       27 

22         28 

44 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

6  ii  17 

23         29 

NATURAL  TANGENTS. 


61 


0 

6' 

12' 

18 

24' 

30 

36 

42' 

48 

54' 

123 

4    5 

45° 

1.  0000 

0035 

0070 

0105 

0141 

0176 

0212 

0247 

0283 

0319 

6    12    18 

24  30 

46 
47 
48 

1-0355 
1.0724 

1.1106 

0392 
0761 
H45 

0428 
0799 
1184 

0464 
0837 
1224 

0501 

0875 
1263 

0538 
0913 
1303 

0575 
095  1 
1343 

0612 
0990 
1383 

0649 
1028 
1423 

0686 
1067 
1463 

6    12    18 
6   J3    19 
7    13    20 

25  3' 
25  32 
26  33 

49 
50 
51 

1.1504 
1.1918 
1-2349 

1544 
1960 
2393 

1585 

2OO2 

2437 

1626 
2045 
2482 

1667 
2088 
2527 

1708 
2131 
2572 

I75C 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 
2753 

7    »4    21 
7    H    *•> 
8    15    23 

28  34 
29  36 
30  38 

52 
53 
54 
55 

1.2799 
1.3270 
1-3764 

2846 
3319 
3814 

2892 

3367 
3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
3514 
4019 

3079 
3564 
4071 

3127 
3613 
4124 

3175 
3663 
4176 

3222 

3713 
4229 

8    16    23 
8    16    25 
9    17    26 

3«  39 
33  4i 

34  43 

1.4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9    18   27 

36  45 

56 
57 
58 

1.4826 

1-5399 
1.6003 

4882 
5458 
6066 

4938 
5517 
6128 

4994 
5577 
6191 

5051 
5637 
6255 
6909 
7603 
8341 

5108 

5697 
6319 

5166 
5757 
6383 

5224 
5818 
6447 

5282 
5880 
6512 

5340 
5941 
6577 

10    19    29 

10     20     30 
II      21      32 

38  48 
40  50 
43  53 

59 
60 
61 

1.6643 
1.7321 
1.8040 

6709 

7391 
8115 

6775 

7461 
8190 

6842 
7532 
8265 

6977 
7675 
8418 

7045 
7747 
8495 

7"3 
7820 
8572 

7182 

7893 
8650 

725J 
7966 
8728 

ii    23    34 

12     24     36 
13     26     38 

45  56 
48  60 
51  64 

62 
63 
64 

1.8807 
1.9626 
2.0503 

8887 
97" 
0594 

8967 

9797 
0586 

9047 
9883 
0778 

9128 
9970 
0872 

9210 
0057 
0965 

3292 
0145 
1060 

9375 
0233 

H55 

9458 
6323 
1251 

9542 
0413 
1348 

14     27     41 

'5    29    44 
16    31    47 

55  68 
5»  73 
63  78 

65 

2-1445 

1543 

1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

'7    34    5' 

68  85 

66 
67 

68 

2.2460 
2-3559 
2.4751 

2566 
3673 
4876 

2673 

3789 
5002 

2781 
3906 
5129 

2889 
4023 
5257 

2998 
4142 
5386 

3109 
4262 
5517 

3220 
4383 
5649 

3332 
4504 
5782 

3445 
4627 
59'6 

18    37    55 
20   40   60 
22    43    65 

74  92 
79  99 
87  108 

69 
70 
71 

2.6051 

2-7475 
2.9042 

6187 
7625 
9208 

6325 
7776 

9375 

6464 
7929 
9544 

6605 
8083 
9714 

6746 
8239 

9887 

6889 
8397 
6061 

7034 
8556 
0237 

7179 
8716 
0415 

7326 
8878 
0595 

24    47    7' 

26    52    78 
29    58    87 

95  "8 
104  130 
"5  '44 

72 
73 
74 
75 

3-0777 
32709 
3.4874 

0961 
2914 
5105 

1146 
3122 
5339 

1334 
3332 

5576 

1524 
3544 
5816 

1716 

3759 
6059 

1910 

3977 
6305 

2106- 
4197 
6554 

2305 
4420 
6806 

2506 
4646 
7062 

32    64    96 
36   72  108 

41     82  122 

129  161 
144  180 
162  203 

3-7321 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

9812 

46    94  139 

i  86  232 

76 
77 
78 
79 
80 
81 
~82 
83 
84 

4.0108 

4-3315 
4.7046 

0408 
3662 
7453 

0713 
4015 
7867 

IO22 

4374 
8288 

1335 
4737 
8716 

1653 
5107 
9152 

1976 

5483 
9594 

2303 
5864 
0045 

2635 
6252 
0504 

2972 
6646 
0970 

53  107  160 
62  124  1  86 
73  146  219 

214  267 
248  310 
292  365 

5.1446 
5.6713 

1929 
7297 
3859 

2422 
7894 
4596 

2924 
8502 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
0405 

5026 
1066 

8548 

5578 
1742 
9395 

6140 
?1?? 

87  175  262 

35«»  437 

6.313^ 

5350 

7920 

6264 

7-H54 
8.144.3 
9-5I44 

2066 
2636 
9.677 

3002 
3863 
9-845 

3962 
5126 

IO.O2 

4947 
6427 

IO.2O 

5958 
7769 
10.39 

6996 
9!52 
10.58 

8062 

0579 
10.78 

9158 
2052 
10.99 

0285 
3572 

11.20 

Difference  -  col- 
umns  cease  to  be 
useful,   owing    to 
the    rapidity  with 
which    the   value 
of  the  tangent 
changes. 

85 
86 
87 
88 

89 

n-43 

11.66 

11.91 

12.16 

1243 

12.71 

13.00 

13-30 

13-62 

13-95 

14.30 
19.08 
28.64 

14.67 
19.74 
30.14 

15-06 
20.45 
31.82 

15.46 
21.20 
33-6Q 

15.89 
22.02 

35-8o 

16.35 
22.90 
38.19 

16.83 
23.86 
40.92 

17-34 
24.90 
44.07 

17.89 
26.03 
47-74 

18.46 
27.27 
52.08 

57-29 

63.66 

71.62 

81.8 

95-49 

114.6 

143-2 

191.0 

286.5 

573-0 

62 


ANTI-LOGARITHMS 


Hants, 

01234     56789 

PROPORTIONAL  PARTS. 

123 

456 

789 

.00 
.0 
.02 
.03 
.04 

1000  1002  1005  1007  1009 
1023  1026  1028  1030  1033 
1047  1050  1052  1054  1057 
1072  1074  1076  1079  1081 
1096  1099  1102  1104  1107 

1012  1014  1016  1019  1021 
1035  1033  1040  1042  1045 
1059  1062  1064  1067  1069 
1084  1086  1089  1091  1094 
1109  1112  1114  1117  1119 

0  0  1 
001 
0  0  1 

1  1 
1  1 
1  1 

222 
222 
222 

0  1  1 

1  1 

222 

.05 
.06 
.07 
.08 
.09 

1122  1126  1127  1130  1132 
1148  1151  1153  1156  1159 
1175  1178  1180  1183  1186 
1202  1205  1208  1211  1213 
1230  1233  1236  1239  1242 

1135  1138  1140  1143  1146 
1161  1164  1167  1169  1172 
1189  1191  1194  1197  1199 
1216  1219  1222  1225  1227 
1245  1247  1250  1253  1256 

Oil 
0  1  1 
Oil 
0  1  1 
0  1  1 

1  1 
1  1 
1  1 
112 
112 

222 
222 
222 
223 
223 

.10 

.11 
.12 

1 

1259  1262  1265  1268  1271 
1288  1291  1294  1297  1300 
1318  1321  1324  1327  1330 
1349  1352  1355  1358  1361 
1380  1384  1387  1390  1393 

1274  1276  1279  1282  1285 
1303  1306  1309  1312  1315 
1334  1337  1340  1343  1346 
1365  1368  1371  1374  1377 
1396  1400  1403  1406  1409 

0  1  1 
Oil 
0  1  1 
Oil 
0  1  1 

113 
122 
122 
122 
122 

228 
223 
223 
233 
233 

'.li 
.17 
.18 
.19 

1413  1416  1419  1422  1426 
1445  1449  1452  1455  1459 
1479  1483  1486  1489  1493 
1514  1517  1521  1524  1528 
1549  1552  1556  1560  1563 

1429  1432  1435  1439  1442 
1462  1466  1469  1472  1476 
1496  1500  1503  1507  1510 
1531  1535  1538  1542  1545 
1567  1570  1574  1578  1581 

Oil 
0  1  1 
Oil 
0  1  1 
Oil 

122 
122 
1  2  2 
122 

233 
233 
233 
233 

•20 
.21 
.22 
.23 
.24 

1585  1589  1592  1596  1600 
1622  1626  1629  1633  1637 
1660,  1663  1667  1671  1675 
legs'  1702  1706  1710  1714 
1738  1742  1746  1750  1754 

1603  1607  1611  1614  1618 
1641  1644  1648  1652  1656 
1679  1683  1687  1690  1694 
1718  1722  1726  1730  1734 
1758  1762  1766  1770  1774 

0  1  1 
Oil 
0  1  1 
Oil 
0  1  1 

222 
222 

333 
333 

222 

334 

.25 

.26 
.27 
.28 
.29 

1778  1782  1786  1791  1795 
1820  1824  1828  1832  1837 
1862  1866  1871  1875  1879 
1905  1910  1914  1919  1923 
1950  1954  1959  1963  1968 

1799  1803  1807  1811  1816 
1841  1845  1849  1854  1858 
1884  1888  1892  1897  1901 
1928  1932  1936  1941  1945 
1972  1977  1982  1986  1991 

Oil 
0  1  1 

Oil 
0  1  1 

.30 
.31 
.32 
.63 
.84 

1995  2000  2004  2009  2014 
2042  2046  2051  2056  2061 
2089  2094  2099  2104  2109 
2138  2143  2148  2153  2158 
2188  2193  2198  2203  2208 

2018  2023  2028  2032  2037 
2065  2070  2075  2080  2084 
2113  2118  2123  2128  2133 
2163  2168  2173  2178  2183 
2213  2218  2223  2228  2234 

Oil 

]  1 
1  1 

223 

344 

223 

344 

1  2 

.35 
.36 
.87 
.38 
.39 

2239  2244  2249  2254  2259 
2291  2296  2301  2307  2312 
2344  2350  2355  2360  2366 
2399  2404  2410  2415  2421 
2455  2460  2466  2472  2477 

2265  2270  2275  2280  2286 
2317  2323  2328  2333  2339 
2371  2377  2382  2388  2393 
2427  2432  2438  2443  2449 
2483  2489  2495  2500  2506 

1  2 

1  2 
1  2 

.40 
.41 
.42 
.43 
.44 

2512  2518  2523  2529  2535 
2570  2576  2582  2588  2594 
2630  2636  2642  2649  2655 
2692  2698  2704  2710  2716 
2754  2761  2767  2773  2780 

2541  2547  2553  2559  2564 
2600  2606  2612  2618  2624 
2661  2667  2673  2679  2685 
2723  2729  2735  2742  2748 
2786  2793  2799  2805  2812 

1  2 
1  2 
1  2 
1  2 
1  2 

234 

455 

334 
834 

456 
456 

45 

.46 
.47 
.48 
.49 

2818  2825  2831  2838  2844 
2884  2891  2897  2904  2911 
2951  2958  2965  2972  2979 
020  3027  3034  3041  3048 
090  3097  3105  3112  3119 

2851  2858  2864  2871  2877 
2917  2924  2931  2938  2944 
2985  2992  2999  3006  3013 
3055  3062  3069  3076  3083 
3126  3133  3141  3148  3155 

1  2 
1  2 
1  2 
1  2 
1  2 

334 
334 
344 
344 

556 
556 
666 
566 

ANTI-LOGARITHMS 


63 


Hants. 

01234     56789 

PROPORTIONAL  PARTS. 

J.  2  3 

456 

789 

.50 
.51 
.52 
.53 
.54 

3162  3170  3177  3184  3192 
3236  3243  3251  3258  3266 
3311  3319  3327  3334  3342 
3388  3396  3404  3412  3420 
3467  3475  3483  3491  3499 

3199  3206  3214  3221  3228 
3273  3281  3289  3296  3304 
3350  3357  3365  3373  3381 
3428  3436  3443  3451  3459 
3508  3516  3524  3532  3540 

112 
1  2  2 
122 
122 
122 

.55 
.56 
.57 
.58 
.59 

3548  3556  3565  3573  3581 
3631  3639  3648  3656  3664 
3715  3724  3733  3741  3750 
3802  3811  3819  3828  3837 
3890  3899  3908  3917  3926 

8589  3597  3606  3614  3622 
3673  3681  3690  3698  3707 
3758  3767  3776  3784  3793 
3846  3855  3864  3873  3882 
3936  3945  3954  3963  3972 

677 

123 

.60 

.61 
.62 
.63 
.64 

3981  3990  3999  4009  4018 
4074  4083  4093  4102  4111 
4169  4178  4188  4198  4207 
4266  4276  4285  4295  4305 
4365  4375  4385  4395  4406 

4027  4036  4046  4055  4064 
4121  4130  4140  4150  4159 
4217  4227  4236  4246  4256 
4315  4325  4335  4345  4355 
4416  4426  4436  4446  4457 

128 

128 

.65 

.66 
.67 
.68 
.69 

4467  4477  4487  4498  4508 
4571  4581  4592  4603  4613 
4677  4688  4699  4710  4721 
4786  4797  4808  4819  4831 
4898  4909  4920  4932  4943 

4519  4529  4539  4550  4560 
4624  4634  4645  4656  4667 
4732  4742  4753  4764  4775 
4842  4853  4864  4875  4887 
4955  4966  4977  4989  5000 

128 

467 

8  9  10 
8  9  10 

.70 

.71 
.72 
.73 
.74 

6012  5023  5035  5047  5058 
5129  5140  5152  5164  5176 
5248  5260  5272  5284  5297 
5370  5383  5395  5408  5420 
5495  5508  5521  5534  5546 

6070  5082  5093  6105  5117 
5188  5200  5212  5224  5236 
6309  5321  5333  5346  5358 
5433  5445  5458  5470  6483 
6659  5572  6585  6598  6610 

8  10  11 

9  10  11 
9  10  12 

.75 
.76 
.77 
.78 
.79 

5623  5636  5649  5662  5675 
5754  5768  5781  5794  5808 
5888  5902  5916  5929  5943 
6026  6039  6053  6067  6081 
6166  6180  6194  6209  6223 

6689  5702  5715  5728  6741 
5821  6834  6848  5861  5875 
6957  5970  5984  5998  6012 
6095  6109  6124  6138  6152 
6237  6252  6266  6281  6295 

9  10  12 
9  11  12 
10  11  12 
10  11  18 
10  11  18 

.80 
.81 
.82 
.83 
.84 

6310  6324  6339  6353  6368 
6457  6471  6486  6501  6516 
6607  6622  6637  6653  6668 
6761  6776  6792  6808  6823 
6918  6934  6950  6966  6982 

6383  6397  6412  8427  6442 
6531  6546  6561  6577  6592 
6683  6699  6714  6730  6745 
6839  6855  6871  6887  6902 
6998  7016  7031  7047  7063 

10  12  18 
11  12  14 
11  12  14 
11  13  14 
11  18  10 

.85 
.86 
.87 
.88 
.89 

7079  7096  7112  7129  7145 
7244  7261  7278  7295  7811 
7413  7430  7447  7464  7482 
7586  7603  7621  7638  7656 
7762  7780  7798  7816  7834 

7161  7178  7194  7211  7228 
7328  7345  7362  7379  7396 
7499  7516  7534  7551  7568 
7674  *«91  7709  7727  7745 
7852  7870  7889  7907  7925 

8 
3 
8 

4 
4 

7  8  10 
7  8  10 
7  9  10 
7  9  11 
7  911 

12  18  15 
12  13  16 
12  14  16 
12  14  16 
18  14  16 

.90 

.91 
.92 
.93 
.94 

7943  7962  7980  7998  8017 
8128  8147  8166  8185  8204 
8318  8337  8356  8375  8395 
8511  8531  8651  8570  8590 
8710  8730  8750  8770  8790 

8035  8054  8072  8091  8110 
8222  8241  8260  8279  8299 
8414  8433  8453  8472  8492 
8610  8630  8650  8670  8690 
8810  8831  8851  8872  8892 

4 

4 
4 

4 
4 

7  9  11 
9  11 
10  12 
10  12 
10  12 

18  10  17 
13  16  17 
14  15  17 
14  16  18 
14  16  18 

.95 

.96 
.97 
.98 
.99 

8913  8933  8954  8974  8995 
9120  9141  9162  9183  9204 
9333  9354  9376  9397  9419 
9550  9572  9594  9616  9638 
9772  9795  9817  9840  9863 

9016  9036  9057  9078  9099 
9226  9247  9268  9290  9311 
9441  9462  9484  9506  9528 
9661  9683  9705  9727  9750 
9886  9908  9931  9954  9977 

4 
4 
2  4 
2  4 
2  5 

10  12 

11  18 
11  13 
11  13 
11  14 

10  17  10 
15  17  19 
15  17  20 
16*18  20 
16  1820 

64 


LOGARITHMS. 


10 

0 

1 

2 

3 

4 

.5 

6 

7 

8 

9 

123 

456 

789 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

Use  Table  second 
page  following 

11 

12 
13 
14 
15 
16. 
17 
18 
19 
20 

0414 
0792 
1139 

0453 

0828 

"73 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

6719 
1072 
1399 

0755 
1106 

T430 

4811 
3  7  10 
3  6  10 

15  19  23 

14    17    21 

13  16  19 

26  30  34 
24  28  31 
23  26  29 

1461 
1761 
2041 

1492 
1790 
2068 

1523 
1818 
2095 

1553 
1847 

2122 

1584 
1875 
2148 

1614 
1903 
2175 

1644 
!93i 

22OI 

1673 

1959 
2227 

17^3 
1967 

2253 

1732 
2014 
2279 

369 
368 
35   8 

12  15  18 
n  14  17 
ir  13  16 

21    24   27 
2O   22    25 
I  8    21    24 

2304 

2553 
2788 

2330 

2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
2900 

2455 
2695 
2923 

3139 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

5   7 
5  7 
4  7 

IO    12    15 

9  12  14 
9  ii  13 

17    20   22 
l6    19    21 

16  18  20 

3010 

3032 

3054 

3075 

3096 

3Il8 

3160 

3181 

3201 

4  6 

8  ii  13 

15  17  19 

21 
22 
23 

3222 
3424 

3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
37H 

3345 
354i 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

4  6 
4  6 
4  6 

8    IO    12 
8    10    12 

7    9  ii 

14  16  18 
14  IS  »-7 
i3  '5  *7 
12  14  16 

12  14    IS 
II    13    15 

24 
26 
26 

3802 

3979 
41^ 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
42OO 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

4   5 
3   5 
3   5 

7    9  ii 
7    9  10 
7    8  10 

27 
28 

29 

30 

43M 
4472 
4624 

4330 
W87 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 

4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

3   5 
3   5 
3  4 

689 
689 
679 

II    13    14 
II    12    14 
10    12    13 

477i 

1786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3   4 

679 

IO    II    13 

31 
32 
33 

4914 
5051 

5185 

1928 
5065 
5198 

4942 

5079 
5211 

4955 
5092 
5224 

4969 
5105 

5237 

4983 
5"9 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

3  4 
3  4 
3  4 

678 

5    7    8 
568 

IO    II     12 
9    11    12 

9  10  12 

34 
35 
36 
37 
38 
39 

5315 
544i 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 

5527 
5647 

54i6 

5539 
5658 

5428 

5551 
5670 

3  4 
4 

4 

568 
5    6    7 
5    6    7 

9  10  ii 
9_io  u 
8  10  ii 

5682 
5798 
59" 

5694 
SSog 
5922 

5705 
5821 

5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 

5999 

5786 

5899 
6010 

3 
3 
3 

5    6    7 
5    6    7 
457 

8    9  10 
8    9  10 
8    9  10 

40 

6021 

6031 

6042 

605.3 

6064 

6075 

6085 

6096 

6107 

6117 

3 

4    5    6 

8    9  10 

41 

42 
43 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 

6325 
6425 

3 
3 
r       3 

456 
4    5    6 
4    5    6 

789 
7     8    9 
7    8    Q 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 
6730 
6821 
6911 

6454 
6551 
6646 

6464 
6561 
6656 

6474 

6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
65-99 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

3 
3 
3 

456 
4    5    6 
4    5    6 

7     8    9 

7     8    9 
7    7    8 

47 
48 
49 

6721 
6812 
6902 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
68& 
£972 

6803 

6893 
6981 

3 

455 

6    7    8 

3 

445 

6    7    8 

50 
51 
52 
63 

6990 

>99S 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

3 

3^   4     5 

6     7    * 

7076 
7160 
7243 

7084 
7168 
7251 

7093 
7177 
7259 

7101 

7185 
7267 

7110 
7193 

7275 

7356 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7M3 
7226 
7308 

7152 
7235 
73i6 

'  2  3 
"V*i  2 

122 

345 
345 
345 

6    7    8 
6    7     7 
667 

54 

7324 

7332 

7340 

7348 

7364 

7372 

738o 

7388 

7396 

122 

345 

667 

LOGARITHMS 


65 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

if)  IVO  l>  CO  IO\  O  H 
if)  |  if)  if)  if)  1  if)  VO  VO 

7404 

7412 

7419 

7427 

7435 

7443 

745  1 

7459 

7466 

7474 

I  2  2 

345 

5  6  7 

7482 
7559 
7634 

7490 
7566 
7642 

7497 
7574 
7649 

7505 
7582 
7657 

7513 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

753<> 
7612 
7686 

7543 
7619 
7694 

755J 
7627 
7701 

I  I  2 

3 

5  6  7 

7709 

7782 
7853 

77i6 
7789 
7860 

7723 
7796 
7868 

773i 
7803 
7875 

7738, 
7810 
7882 

7745 
7818 
7889 

7752 
7825 
7896 

7760 
7832 
7903 

7767 

7839 
7910 

7774 
7846 

79^7 

I  I  2 
I  I  2 
112 

3 
3 
3 

5  6  7 
5  6  t 
5  6  fc 

62 
63 
64 

7924 

7993 
8062 

7931 
Sooo 
8069 

7938 
8007 
8075 

7945 
8014 

8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
805  q 
812? 

I  1  2 
112 
I  I  2 

112 

3  3 
3  3 
3  3 

5  6  6 

5  S  6 

s  s  e 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

3  3 

5  S  <- 

66 
67 
68 

8195 
8261 
8325 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 

8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

8445 
8506 
8567 

I  I  2 
I  I  2 
112 

3  3 
3  3 
3  3 

5  5  6 
5  5  6 
4  5  6 

69 
70 
71 

8388 
845J 
8^13 

8395 
8457 
8519 

8401 
8463 
8S2* 

8407 
8470 

853i 

8414 
8476 

8*37 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 

8555 

8439 
8500 
8561 

112 

*\3 

2  ^ 

4  5  (• 

72 
73 
74 

S573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
874S 

112 
I  I  2 

2   3 
2   3 

455 
455 

75 

875J 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

112 

233 

4  5  5 

76 
77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8876 
8932 

8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8^4 
8910 
8965 

885-9 
8915 
8971 

112 
112 
I  I  2 

233 
233 

4  5 

4  5 

79 
80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
^047 
9101 

8998 

9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9*33 

I  I  2 
112 
112 

233 
233 
233 

4  5 
4  5 
4  5 

82 
83 
84 

9138 
9191 
9243 

9*43 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9l65 
9217 
9269 

9170 
9222 
9274 

9175 
9227 

9279 

9180 
9232 
9284 

9186 
9238 
9289 

I  I  2 

2  3  3j  4  4  5 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 
87 
88 

9345 
9395 
944S 

9350 
9400 
9450 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

0  I  I 

2   2   3 

3  4  4 

89 
90 
91 

9494 
9542 
959° 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

95i8 
9566 
9614 

9523 
957' 
9619 

9528 
9576 
9624 

9533 
958i 
9628 

9538 
9580 
9^33 

Oil 
Oil 
Oil 

2   2   3 

223 
223 

3  4  A 
3  4  A 
344 

92 
93 
94 

9638 
9685 
973' 
9777 

9643 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 
9791 

9657 
9703 
9/50 

9661 
9708 
9754 

9666 
97*3 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

0  I  1 
Oil 
0  I  I 

223 
223 

344 
344 

95 

9782 

9786 

9795 

9800 

9805 

9809 

9814 

9818 

Oil 

223 

3  4  4 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 

9934 

9850 
9894 
9939 

9854 
9899 

9943 

9859 
9903 
9948 

9863 
9908 
9952 

O  I  I 
0  I  I 
0  I  I 

223 
223 
223 

344 
344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

999  1 

9996 

0  I  I 

223 

334 

66 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

00000 

043 

087 

130 

173 

217 

s6o 

303 

346 

389 

101 
102 
103 

432 
860 

01  284 

475 
9°3 
326 

5i8 
945 
368 

561 
988 
410 

604 
030 
452 

647 
072 
494 

689 
US 
536 

732 
157 
578 

775 
199 
620 

817 
242 
662 

104 
105 
106 

703 

02  IIQ 

531 

745 
1  60 
572 

787 
202 
612 

828 
243 
653 

870 
284 
694 

912 
325 

735 

953 
366 
776 

995 
407 
816 

036 
449 

857 

078 
490 
898 

107 
108 
109 

938 
03342 

743 

979 

383 

782 

019 

423 

822 

060 

463 
862 

100 

503 
902 

141 

543 
941 

181 

583 
981 

222 
623 
O2  1 

262 
663 
060 

302 
703 

100 

To  find  the  logarithm  of  a  number:  First,  locate  in  the 
table  the  mantissa  which  lies  in  line  with  the  first  two  figures  of  the 
number  and  underneath  the  third  figure,  then  increase  this  mantissa 
by  an  amount  depending  upon  the  fourth  figure  of  the  number  and 
found  by  means  of  the  interpolation  columns  at  the  right;  secondly, 
determine  the  characteristic,  or  the  exponent  of  that  integer  power 
of  10  which  lies  next  in  value  below  the  number;  for  example, 
log  600=  0.7782  -h  2. ;  log  73.46=  0.8661  +  1. ; 

log  .006-0.7782-3.;  log  .7346=0.8661-1.; 

log  6.003=  0.7784  +  0. ;  log  7349=  0.8662  +  3. 

The  logarithm  of  a  product  of  two  or  more  numbers  is  the  sum  of 
the  logarithms  of  its  factors;  for  example, 

log.  (.0821  X  463.2)  =  (0.9143  -  2.)  +  (0.6658  +  2.)  =  0.5801  +  1. 
The  logarithm  of  a  quotient  is  the  difference  between  the  logar- 
ithms of  the  dividend  and  divisor;  for  example, 

log.  (.5321  -f-  916)  =  (0.7260  - 1 .)  -  (0.9619  +  2.)  =  0.7641  - 4. 
The  logarithm  of  a  power  or  root  of  a  number  is  the  exponent 
times  the  logarithm  of  the  number;  for  example, 

log  V^63)3=3/2  X  (0.9360— 1.)  =  0.9040  -  1. 

To  find  the  number  from  its  logarithm:  Locate  in  the  table 
the  mantissa  next  less  than  the  given  mantissa,  then  join  the  figure 
standing  above  it  at  the  top  of  the  table  to  the  two  figures  at  the 
extreme  left  on  the  same  line  as  the  mantissa,  and  finally  to  these 
three  join  the  figure  at  the  top  of  the  interpolation  column  which 
contains  the  difference  between  the  two  mantissae.  In  the  four- 
figure  number  thus  found,  so  place  the  decimal  point  that  the 
number  shall  be  the  product  of  some  number,  that  lies  between 
1  and  10,  by  a  power  of  10  whose  exponent  is  the  characteristic 
of  the  logarithm.  For  example, 

antilog  (0.6440 +  3)  =  4405; 
antilog  (0.3069  -  2)  =  .02027. 

Caution.  In  adding  and  subtracting  logarithms  it  is  well  to 
remember  that  the  mantissa  is  always  essentially  positive  and  may 
or  may  not  therefore,  have  the  same  sign  as  its  characteristic. 


INDEX 


Absolute,  humidity            -  41 

Acceleration,  uniform       -  5 

due  to  gravity       -          -  10 

apparatus,  described     -  6 

normal                              -  8 

Adiabatic  changes,  law  for  41 

Air  thermometer,  constant 

volume                              -  37 

Auguste's    Psychrometer  42 

Boiling  Point,  correction  -  27 
Boiling     Point,     variation 

with  pressure         -  35 

Boyle's  Law,  used    -         -  45 
Calibration 

absolute,  of  thermometer  25 

plot  of                           -  28 

sample  set  of  data     -  30 

of  thermometer  tube     -  26 

of  set  of  weights  -  23 

Centripetal  force      -         -  8 

Charles'  Law,  tested         -  38 

Clement    &    Desormes, 

method  of  -  -  44 
Coincidences,  method  of  12 
Coefficient  of  expansion  of 
air,  with  air  thermome- 
ter -  -  37 
Dew  point,  defined  -  41 
Efflux  of  gases  21 
Force,  centripetal  -  -  8 
Freezing  Point,  correction  27 
Fusion,  heat  of,  of  tin  -  46 
"G"  determination  of, 

with    fall- machine          -  5 
with  pendulum      -         -  10 
Heat,  mechanical  equiva- 
lent of           ...  18 


Heat  of  Neutralization    -  33 

Heat  of  Solution  31 

Humidity,     absolute,     de- 
fined    -                             -  41 
relative,  defined   -         -  41 

Hygrometer,    formula   for 

use  with        -                   -  43 

Alluard's  42 

wet-and-dry-bulb           -  42 

Mechanical  Equivalent  of 

Heat,  defined                   -  18 

by  Calendar's  method  18 

Moment   of   inertia    of    a 

disk  15 

Melting  points,  zinc,  tin  -  46 

Momentum      -  14 

Motion,  study  of  uni- 
formly accelerated         -  5 

Normal  Solution       -         -  34 

Pendulum,  simple     -  12 

Period,     of    vibration     of 

pendulum     -                   -  10 

Ratio  of  specific  heats      -  44 

Specific  heat,  of  graphite, 

porcelain,  fire  clay         -  48 

Thermometer,     constant 

volume  air    -  37 

Uniform,  accelerated  mo- 
tion   .  -         -         -         -  5 
circular  motion     -         -  8 

Vapor  pressure  and  tem- 
perature       -                   -  39 

Velocity,  average  7 

Wet-    and    dry-bulb,    hy- 
grometer      *         -         -  42 


PHYSICAL   MEASUREMENTS 


MINOR 


PART  III.     MAGNETISM  AND  ELECTRICITY 
1916-1917 


PHYSICAL   MEASUREMENTS 


WETZEL  BROS.  PRINTING  CO. 

2110  ADDISON  STREET 

BERKELEY,  CAL. 


PHYSICAL   MEASUREMENTS 

A  Laboratory  Manual  in  General  Physics 
For  Colleges 

by 
RALPH  S.  MINOR,  PH.  D. 

Associate   Professor  of  Physics,    University  of  California 


IN  FOUR  PARTS 

PART  III. 
MAGNETISM  AND  ELECTRICITY 


Edited  in  Collaboration  With 
RAYMOND  B.  ABBOTT,  M.  S. 

Instructor  in  Physics,    University  of  California 


Berkeley,  California 
1916 


Copyrighted  in  the  year  1916 

by 
Ralph  S.  Minor 


LIST  OF  EXPERIMENTS 

Magnetism  Page 

General  Statements  and  Directions   -  -       7 

51.  Plotting  of  Magnetic  and  Electric  Fields  9 

52.  Laws  of  Magnetic  Force    -  -     12 

53.  Relative  Determination  of  H  14 

54.  Absolute  Determination  of  H    -  -     15 

55.  Magnetic  Induction 

B  and  H  Curve       -  19 

Electricity 

General  Directions    -  -     23 

Theory  of  the  Tangent  Galvanometer  -     23 

61.  Test  of  the  Tangent  Law  24 

62.  Reduction  Factor  of  Galvanometer 

By  Deposition  of  Copper  -     26 

63.  Measurement  of  Resistance 

Wheatstone  Bridge  -     29 

64.  Comparison  of  Resistances 

Carey  Foster's  Method  -     30 

65.  Absolute  Determination  of  Resistance 

Calorimeter  Method  -     32 

66.  Electromotive  Force 

PoggendorfTs  Method.    The  Potentiometer     -     34 

67.  Thermo-Electromotive  Force 

Calibration  of  a  Galvanometer  as  a  Voltmeter     37 

68.  The  Earth-Inductor 

Relative  Calibration  of  a  Galvanometer  -     40 

69.  Resistance  of  an  Electrolyte 

Kohlrausch's  Method.     Alternating  Current- 
Telephone       -  -     42 

70.  Study  of    Polarization    Effects   in    a    Leclanche 

Cell 

Galvanometer  used  as  a  Voltmeter  -     45 

71.  Comparison  of  Capacities  -     46 

72.  Efficiency  of  Electrical  Heating  Devices    -         -     47 


REFERENCES 

Brooks  and  Poyser:     Electricity  and  Magnetism. 
Duff:     Textbook  of  Physics  (Fourth  Edition.) 
Ganot:     Textbook  of  Physics   (18th   Edition). 
Kimball:     College  Physics. 
Thompson:     Electricity  and  Magnetism. 

Kaye  and  Laby:     Physical  and  Chemical  Constants. 
I/andolt  and  Bernstein:     Physical  and  Chemical  Tables. 
Smithsonian  Institute:     Physical  Tables. 


MAGNETISM 

General  Statements  and  Directions. 

The  phenomena  of  magnetism  and  current  electricity 
are  almost  inseparably  connected.  A  magnetic  field  of 
force  always  exists  in  the  medium  surrounding  a  conduc- 
tor carrying  a  current  of  electricity,  and  if  a  conductor 
be  moved  across  a  magnetic  field  an  electromotive  force  is 
always  induced  in  the  conductor.  Whether  a  magnetic 
field  be  due  to  the  magnetism  of  the  earth,  to  a  system  of 
magnets,  or  to  electric  currents  in  conductors,  the  field  at 
any  point  is  completely  determined  when  the  direction 
and  intensity  of  the  force  at  the  point  are  known. 

1.  The  direction  of  the  magnetic  force  is  arbitrarily  de- 
fined as  the  direction  in  which  an  isolated  north-seeking 
magnet  pole  would  be  urged  by  the  magnetic  force  if  placed 
in  a  magnetic  field.    Imaginary  lines  showing  at  all  points 
the  direction  in  which  the  magnetic  force  acts  are  called 
lines  of  force.     Magnetic  lines  of  force  are  closed  curves, 
i.  e.,  they  have  no  free  ends. 

2.  The  strength  or  intensity  of  the  magnetic  force   at 
any  point  in  a  magnetic  field  is  numerically  equal  to  the 
force  in  dynes  which  would  be  exerted  by  the  field  upon  a 
unit  magnetic  pole  placed  at   that  point,  a  unit  magnetic 
pole  being  defined  as  a  magnetic  pole  of  such  strength 
that  when  placed  in  air  at  a  distance  of  one  centimeter 
from  a  similar  pole  it  will  repel  it  with  a  force  of  one  dyne. 
Dimensionally  the  intensity  of  the  magnetic  field  is  force 
divided   by  pole  strength.     The  unit  of   magnetic  field 
intensity  is  called  the  gauss. 

The  intensity  of  the  earth's  magnetic  field  is  usually 
represented  by  the  letter  H,  field  intensity  in  general  will 
be  designated  by  H.  This  might  be  expressed  by  saying 


8  MAGNETISM 

that  the  magnetic  force  would  produce  H  lines  per  square 
centimeter  in  air. 

3.  If  a  piece  of  iron  or  other  magnetizable  substance 
be  placed  in  the   magnetic  field  some   of  the   magnetic 
lines  pass  through  the  substance   and   magnetize  it;   in 
fact,  more  lines  pass  through  iron  than  through  the  air 
which  it  displaces.     The  actual  number  of  magnetic  lines 
that  run  through  unit  area  of  cross-section  in  the  iron 
or  other  material  is  denoted  by  the  letter  B — and  is  a 
measure  of  the  magnetization. 

The  ratio  of  B  to  H  is  called  the  permeability,  /x. 

4.  The  poles  of  a  magnet  are  sometimes  thought  of 
as  points,  usually  near  the  ends  of  a  magnet,  at  which 
the  magnetic  charge  may  be  regarded  as  concentrated. 
In  any  given  magnet,  however,  the  magnetization  is  not 
concentrated  in  two  points,  but  is  distributed  over  the 
magnet.    A  more  extended  conception  of  a  magnetic  pole 
may  be  obtained  by  considering  a  magnet  placed  in  a 
uniform  magnetic  field.     Then  the  forces  acting  on  the 
elementary  positive  magnetic  charges  will  be  a  system  of 
parallel  forces  all  acting  in  the  same  direction.     By  com- 
position of  forces  these  parallel  forces  may  be  replaced  by 
a  single  force  acting  at  a  point  TV  which  is  the  center  for 
this  system  of  parallel  forces.    This  point,  N,  is  the  posi- 
tive or  north-seeking  pole  of  the  magnet.     In  the  same 
way  the  forces  acting  on  the  elementary  negative  charges 
may  be  replaced  by  a  single  resultant  force  acting  at  5, 
the  negative  pole  of  the  magnet. 

If  in  the  above  case  the  strength  of  the  uniform  field  is 
of  unit  intensity  then  the  magnitude  of  the  force  acting 
at  N  or  5  is  equal  numerically  to  the  pole  strength  of 
the  magnet. 


51]  MAPPING  MAGNETIC  LINES  OF  FORCE  9 

If  the  field  is  not  uniform  the  resultant  of  all  the  forces 
acting  will  not  pass  through  the  points  N  or  S,  so  that  as 
far  as  the  magnet  is  concerned  its  poles  are  a  convenient 
but  highly  idealized  conception. 

5.  The  magnetic  moment  of  a  magnet,  however,  is  an 
important    quantity    with    a    perfectly    definite    physical 
meaning.     It  is  found  that  the  tendency  of  a  magnet  to 
turn  or  to  be  turned  by  another  magnet  depends  not  only 
upon  the  strength  of  its  poles,  m,  but  also  upon  their 
distance  apart,  21.     The  product  of  these  two  quantities 
mx2l  is  called  the  magnetic  moment,  M,  of  the  magnet. 
If  the  magnet  be  placed  in  a  field  of  intensity,  H,  and  the 
turning   moment    (G)    be   found,    when   the   axis   of   the 
magnet  is  at  right  angles  to  the  lines  of  force,  then 

M  H  =  G. 

6.  In  handling  magnets  care  must  be  taken  not  to 
bring  the  magnets  into  strong  magnetic  fields,  or  to  jar 
them.     Rough  treatment  may  change  the  strength  of  the 
magnet  and  this  change  in  the  middle  of  an  exercise  will 
necessitate  the  repetition  of  all  previous  measurements 
involving  the  magnetic  moment  of  the  magnet.     Always 
protect  the  jewel  of  the  compass  needle,  never  move  nor 
leave  the  compass  without  first  releasing  the  needle  from 
the  pivot. 

51.     PLOTTING  OF  MAGNETIC  AND  ELECTRIC  FIELDS. 

Equipotential  Lines. 
A.     Magnetic  Field. 

The  object  of  this  part  of  the  exercise  is  to  plot  the  magnetic 
field  due  to  the  combination  of  the  fields  of  two  magnets 
with  the  earth's  magnetic  field. 

(a)  Fasten  two  Robison  spherical-ended  magnets  with 
their  axes  vertical  to  the  underside  of  the  laboratory 


10  PLOTTING  AN  ELECTRIC  FIELD  [51 

table.  The  magnets  should  be  about  20  centimeters 
apart  and  the  line  through  them  should  coincide  with 
the  magnetic  meridian.  Upon  the  top  of  the  table,  over 
the  magnets,  fasten  a  large  sheet  of  paper.  To  trace  a 
force  line,  place  the  compass  needle,  which  should  not 
be  over  2  cm.  long,  on  the  paper  and  make  a  dot  on  the 
paper  under  each  end  of  the  needle.  Now  move  the 
needle  until  the  S-pole  is  over  the  dot  just  made  under 
the  N-pole,  and  make  a  dot  under  the  new  position  of 
the  N-pole.  Continue  this  process  until  you  reach  a 
place  where  the  compass  suddenly  reverses  its  direction, 
or  the  line  runs  off  the  paper;  and  then,  working  in  the 
opposite  direction  from  the  starting  point,  proceed  until 
a  second  such  point  is  found.  In  this  way  plot  four  or 
five  lines  symmetrically  located,  enough  to  show  clearly 
the  complete  outline  of  the  field. 

Plot  with  especial  care  the  ends  of  the  lines  near  the 
singular  points,  i.  e.,  points  in  which  the  needle  will 
assume  no  definite  direction. 


(6)  Using  red  ink  draw  a  second  series  of  lines  which 
are  everywhere  approximately  perpendicular  to  the  first 
set.  These  then  are  the  lines  of  equipotential. 

B,     Electric  Field. 

The  object  of  this  part  of  the  exercise  is  to  plot  the  equipo- 
tential lines  in  an  electric  field  and  to  draw  the  electric  force 
lines. 

(c)  Fill  the  glass  bottomed  tray  to  a  depth  of  three 
or  four  millimeters  with  a  solution  of  ammonium  chloride. 
Connect  one  cell  of  storage  battery  to  the  primary  of  the 
small  induction  coil  and  then  connect  the  secondary 
terminals  of  the  coil  with  two  electrodes  A  and  B  which 


51]  PLOTTING  AN  ELECTRIC  FIELD  11 

have  their  pointed  ends  turned  downward  so  as  to  dip 
into  the  solution.  These  electrodes  should  be  about  20 
centimeters  apart. 

When  the  coil  is  in  operation  the  points  A  and  B  will 
be  maintained  at  different  potentials,  and  although  this 
is  a  constantly  varying  potential  difference  the  equipoten- 
tial  lines  will  preserve  the  same  configuration,  although, 
of  course,  assuming  different  absolute  values.  This  fact 
enables  us  to  use  a  telephone  receiver  in  exploring  the 
equipotential  lines.  For  if  both  of  its  terminals  lie  on  the 
same  equipotential  surface,  no  current  will  pass  through 
the  receiver.  If,  however,  the  two  terminals  lie  on  sur- 
faces of  different  potentials,  a  charge  is  urged  through 
the  receiver  from  the  point  of  high  potential  to  that  of 
low  potential  and  the  receiver  responds  by  buzzing. 

Connect  the  telephone  receiver  to  one  fixed  electrode 
and  one  exploring  electrode  furnished  with  a  handle. 
Place  the  fixed  electrode  about  three  centimeters  from 
the  electrode  A  on  the  line  A  B  and  with  the  exploring 
electrode  find  enough  points  for  which  there  is  no  buzzing 
in  the  receiver  so  that  the  equipotential  line  through  the 
position  of  the  fixed  electrode  can  be  plotted. 

Move  the  fixed  electrode  about  three  centimeters 
toward  B  and  locate  a  second  equipotential  line,  and 
continue  in  this  way  until  four  or  five  lines  covering  the 
entire  field  have  been  explored. 


(d)  Plot  on  co-ordinate  paper  to  one  half  scale  the 
lines  so  found.  The  lines  of  force  will  form  a  system 
everywhere  at  right  angles  to  these  equipotential  lines. 
Draw  this  system  in  red  ink. 


12  LAWS  OF  MAGNETIC  FORCE  [52 


52.     LAWS  OF  MAGNETIC  FORGE. 

In  every  real  magnet  there  is  always  an  equal  quantity 
of  both  kinds  of  magnetism.  It  is  impossible  to  isolate  a 
region  of  positive  magnetism ;  if  the  magnet  be  broken 
each  piece  is  found  still  possessed  of  equal  quantities 
of  both  kinds  of  magnetism.  If  we  have  a  bar  magnet 
the  free  magnetism  is  strongest  in  a  region  near  the  end. 
In  short,  thick  magnets  this  region  is  comparatively 
large;  as  the  length  increases  and  the  diameter  decreases 
the  free  magnetism  is  found  nearer  the  end.  In  a  very 
long  thin  filament  the  free  magnetism  will  be  at  the  end. 

The  object  of  this  experiment  is  to  study  the  variation  in 
field  intensity  with  distance  in  the  magnetic  field  near  one  end 
of  a  long  magnet  and  in  a  plane  perpendicular  to  its  axis. 

The  magnet  is  a  piece  of  steel  ribbon  about  150  centi- 
meters long.  To  avoid  the  formation  of  consequent  poles 
it  should  be  magnetized  by  placing  it  inside  a  long  coil  of 
wire  carrying  a  heavy  current,  the  magnetic  circuit  being 
completed  outside  the  coil,  from  one  end  of  the  magnet  to 
the  other,  by  means  of  pieces  of  soft  iron. 

The  compass  needle  is  a  magnetized  steel  disk,  20  mil- 
limeters in  diameter,  with  a  long  aluminum  pointer  set 
at  right  angles  to  the  magnetic  axis  of  the  disk. 

The  magnet  should  be  mounted  with  its  axis  vertical 
and  one  end  on  a  level  with  the  compass  needle. 

(a)  With  the  magnet  removed  to  a  considerable  dis- 
tance the  compass  should  first  be  set  so  that  the  pointer 
reads  zero  and  then  leveled  so  that  the  pointer  clears  the 
scale. 


52]  LAWS  OF  MAGNETIC  FORCE  13 

Place  £he  magnet  on  the  east-and-west  line,  west  of  the 
compassjlneedle,  and  read  the  deflections  produced  for 
the  series  of  distances  20,  21,  22,  28,  29,  30,  centimeters 
between  the  compass  needle  and  the  magnet.  Readings 
should  be  taken  from  both  ends  of  the  pointer. 

(b)  Repeat  the  readings  for  the  same  distances  to 
the  east  of  the  compass  needle. 


(c)  Show  that  if  the  compass  needle  is  deflected  by 
a  horizontal  force  acting  in  an  east-and-west  direction, 
the  magnitude  of  the  force  is  proportional  to  the  tangent 
of  the  angle  of  deflection. 

State  the  law  for  the  mutual  action  of  magnet  poles. 

If  the  magnetic  force  varies  as  some  power,  n,  of  the 
distance  the  product  dntan  a  should  be  a  constant.  That 
is,  d22n  tan  a22  =  k  =  d2sn  tan  a28  etc.  From  which  we  get 
d28n  /  d22a  =  tan  a22  /  tan  a2S ,  and  n  =  (log  tan  a^  -  log  tan 
a2s)  I  (log  d2s  -  log  dv). 

Average  the  readings  for  each  distance.  By  pairing 
each  of  the  first  three  values  of  dntan  a  with  each  of  the 
last  three  in  turn,  calculate  nine  values  for  n  and  take 
their  average.  Find  the  average  difference  in  percent 
between  these  values  and  your  result. 

What  error  is  eliminated  by  reading  both  ends  of  the 
pointer? — by  taking  readings  both  east  and  west  of  the 
needle? 

X 

(d)  Two  assumptions  have  been  made  whose  effect 
should  be  considered.  The  force  due  to  one  pole  has  been 
neglected  when  considering  that  of  the  other.  Calculate 
the  force  discarded,  assuming  the  inverse  square  law, 
(n  =  2)  and  express  it  in  per  cent  of  the  force  used,  for 


14  RELATIVE  DETERMINATION  OF  H  [52-53 

the  largest  and  smallest  values  of  the  distance,  d.  Second- 
ly, the  force  lines  from  the  magnet  pole  to  compass  needle 
have  been  assumed  parallel.  Calculate  the  error  this 
produces,  in  per  cent  of  the  force  used,  for  the  same 
distances,  the  distance  between  the  poles  of  the  needle 
being  taken  as  18  millimeters  and  its  direction  being 
assumed  perpendicular  to  the  east-and-west  line. 

53.     RELATIVE  DETERMINATION  OF  H. 
A.     Method  of  Oscillations. 

If  a  magnet  vibrating  as  a  torsion  pendulum,  at  a  point 
where  the  horizontal  intensity  has  the  value  Hi,  has  a 
period  Ti,  and  at  a  second  point  the  period  is  found  to  be 
T2,  then  the  horizontal  intensity,  H2,  at  the  second  point 
is  given  by  the  relation  Hi  :  H2  =  TV  :  TV. 

(a)  Determine  the  value  of  H,  using  the  method  of 
oscillations.  The  amplitude  of  the  vibration  should  be 
small  and  care  taken  to  prevent  the  magnet  from  swinging. 

B.  Using  a  Tangent  Galvanometer,  an  Ammeter  and  an 
adjustable  Resistance.* 

Set  up  the  tangent  galvanometer  at  the  point  where 
the  value  of  H  is  to  be  found  and  connect  in  series  with  it 
an  ammeter,  a  single  storage  or  Daniell  cell,  and  an  ad- 
justable resistance.  The  ammeter  should  be  placed  at 
some  distance  from  the  galvanometer. 

Adjust  the  resistance  until  the  galvanometer  reads  the 
angle  whose  tangent  is  equal  to  one  tenth  of  the  galvanometer 
constant,  the  ammeter  reading  will  then  give  H  directly. 


*See  R,  B.  Abbott,  School  Science  and  Mathematics,  Vol.  XII, 
page  533,  1913. 


53-54]  ABSOLUTE  DETERMINATION  OF  H  15 

10    TJ 

Since  in  general     I  =  -  tan  a,    where   n  is    the    number  of 
2-rrn/r 

turns  and  r  the   mean  radius  of  the  coils   (see  Theory  of  Tangent 
Galvanometer,  p.  24).       If    then,  we    so    adjust    the  current  that 


tan  a    =  -  X  -     it  follows  by  substitution 
10  r 

I  =  -        -X--X  lirn/r  =  H,  or  I  =  H  numerically. 
lirn/r         10 

54.     ABSOLUTE  DETERMINATION  OF  H. 

The  following  absolute  determination  of  the  horizontal 
intensity  of  H  of  the  earth's  magnetic  field  at  any  point 
requires  the  measurement  of  the  values  of  the  quotient 
M/  H  and  the  product  M  H,  where  M  is  the  moment  of 
the  magnet. 

If  a  magnet  be  placed  with  its  axis  at  right  angles 
to  the  meridian,  from  the  observed  deflection  of  a  com- 
pass needle  or  magnetometer  placed  on  the  axis  of  the 
magnet  the  quotient  M  /  H  may  be  calculated. 

If  this  same  magnet  be  put  in  a  bifilar  suspension  with 
the  plane  of  the  wires  at  right  angles  to  the  meridian, 
from  the  deflection  produced  by  the  earth's  field  in  turn- 
ing the  magnet  toward  the  magnetic  meridian  the  prod- 
uct M  H  may  be  found.  From  these  two  quantities  either 
H  or  M  may  be  found.  Either  method  may  be  used  alone 
to  compare  the  moments  of  two  magnets. 

The  magnet  used  must  be  kept  away  from  other  mag- 
nets or  magnetic  bodies  and  be  carefully  handled  until 
both  experiments  have  been  completed  so  as  not  to  change 
the  amount  or  distribution  of  the  magnetism. 

A.     Finding  M/H  by  Deflection  in  End-on  Position. 

(a)  As  deflecting  magnet,  one  of  the  Robison  type  will 
be  used  since  these  have  practically  true  poles,  located  at 


16  ABSOLUTE  DETERMINATION  OF  H  [54 

the  centers  of  the  spherical  ends.     Measure  the  distance, 
21,  between  the  centers  of  the  spheres. 

(b)  Place  a  magnetometer  at  the  point  at  which  H  is 
to  be  measured  and  place  the  magnet  with  its  axis  on  the 
east-and-west  line,  east  or  west  of  the  compass  at  a  dis- 
tance which  is  relatively  large  compared  with  length  of 
the  magnet  in  the  magnetometer.  Measure  the  distance 
from  the  center  of  the  magnetometer  to  the  nearest  end 
of  the  magnet,  and  read  the  deflection  of  the  magneto- 
meter using  a  telescope  and  scale  placed  one  meter  away. 
Reverse  the  magnet  and  again  note  the  deflection.  Repeat 
the  measurements  with  the  magnet  on  the  other  side  of 
the  magnetometer  and  the  same  distance  away.  With  the 
aid  of  the  data  taken  in  (a)  find  L,  the  mean  distance 
from  the  center  of  the  magnetometer  to  the  point  midway 
between  the  poles  of  the  magnet. 


(c)  Assuming  the  inverse  square  law  derive  an  ex- 
pression for  the  strength  of  field  due  to  a  magnet  at  any 
point,  s,  on  its  axis.  Call  the  distance  between  the  poles 
of  the  magnet  21,  its  pole  strength  m,  and  L  the  distance 
from  the  point  s  to  a  point  midway  between  the  poles. 

Show  that,  if  this  force  acting  in  an  east-and-west 
direction  deflects  a  compass  needle,  the  magnitude  of 
the  force  is  proportional  to  the  tangent  of  the  angle  of 
deflection,  a,  and  if  the  length  of  the  compass  needle  be 
negligible  in  comparison  to  the  distance  L  show  that 

M      (L2-/2)2 
-H'    ~^Ltana 

where  2  /  m  =  M  =  the  moment  of  the  magnet. 

Using  this  relation  and  the  above  data  calculate  the 
numerical  value  of  M  /  H. 


54]  ABSOLUTE  DETERMINATION  OF  H  17 

B.     Finding  M  H  by  the  bifilar  method. 

(a)  Suspend    the    carriage    for    the    magnet    by    two 
lengths  of  fine  braided  fish-line,   adjustable  from  above 
so  that  they  are  in  a  plane  perpendicular  to  the  magnetic 
meridian.     The  vane  extending  from  the  carriage  should 
be  placed  in  a  vessel  of  water  to  "damp"  the  vibrations. 

To  find  the  zero  position  place  a  brass  rod  of  about  the 
same  size  as  the  magnet  in  the  carriage  and  adjust  the 
telescope  and  scale,  placed  about  a  meter  from  the  car- 
riage, until  the  zero  of  the  scale  seen  in  the  telescope 
after  reflection  in  the  mirror  coincides  with  the  cross-hairs. 
Measure  the  distance  from  the  mirror  to  the  scale. 

Remove  the  brass  rod,  place  the  magnet  in  the  carriage, 
and  read  the  deflection.  It  is  not  necessary  to  wait  until 
the  system  comes  to  rest  before  taking  a  reading.  Where 
the  oscillations  are  small  determine  the  resting  point  by 
reading  the  turning  points  as  in  weighing.  Reverse  the 
magnet  and  again  read  the  deflection.  Make  several  read- 
ings, reversing  the  magnet  each  time,  and  take  the  mean. 

(b)  Weigh  the  magnet  and  carriage  on  the  trip  scales 
to  one  decigram.    Measure  the  distance  d  between  the  two 
wires  of  the  bifilar  suspension  and  also  the  length  of  the 
wires  from  the  point  of  suspension  to  the  top  of  the  car- 
riage. 


(c)  Since  the  angle  of  deflection  a  is  small,  A  A'  is 
approximately  perpendicular  to  AB  and  may  be  ex- 
pressed in  terms  of  quantities  measured  in  (a)  A  A'  = 
AO  sin  a.  =  (d/2)  sin  a. 

Assume  that  the  weight  of  the  magnet  and  carriage 
(mg  dynes)  is  equally  divided  between  the  two  wires  of 


18 


ABSOLUTE  DETERMINATION  OF  H 


[54 


the  suspension.     In  the  force  parallelogram,   figure   (c), 
we  see  that  the  nearly  horizontal  restoring  force  F  which 


D 


Fig    1. — Illustrates  Experiment  54.  (c.) 

acts  along  the  line  A  A'  is  the  resultant  of  the  upward 
tension  of  the  string  T  and  half  the  weight  of  the  magnet 
and  carriage  mg/2.  An  equal  force  acts  along  BB'.  The 
restoring  couple  is  thus  = 

d*mg  sin  a 


4CA 

Show  with  the  aid  of  a  diagram  that  the  moment  of 
the  magnetic  deflecting  couple  is  =  H  m  21  cos  a  = 
H  M  cos  a. 

Calculate  the  numerical  value  of  the  product  M  H. 


Combine  the  results  for  M/ H  and  MH  and  find  the 
value  of  H. 


55]  MAGNETIC  INDUCTION  19 

55.     MAGNETIC  INDUCTION. 
B  and  H  Curve. 

The  magnetic  field  intensity  at  the  center  of  a  long 
solenoid  is,  for  air, 

(1)     H  =  4  TT  n  I/Wl 

where  n  is  the  number  of  turns,  I  is  the  current  in  amperes, 
and  /  is  the  length  of  the  solenoid  in  centimeters.  The 
total  number  of  lines  of  force  threading  the  solenoid  is 
H  A,  where  A  is  the  mean  area  of  its  cross  section,  to  be 
found  by  taking  the  average  of  the  cross-sections  of  each 
separate  layer. 

A  secondary  winding  placed  around  the  solenoid  will 
have  the  same  number  of  lines  threading  it  as  pass  through 
the  solenoid. 

A  ballistic  galvanometer  connected  to  the  secondary 
winding  may  be  calibrated  to  indicate  the  change  in  the 
number  of  lines  of  force  which  thread  the  coil,  by  the 
amount  of  its  deflection  when  a  current  /  is  suddenly  sent 
through  the  primary  winding. 

Calibration  of  a  Ballistic  Galvanometer. 

(a)  The  solenoid  is  mounted  on  a  base  provided  with 
a  commutator,  a  key  K,  and  four  knife  switches,  num- 
bered 1,  2,  3  and  4  for  changing  the  resistance  in  the  cir- 
cuit. Connect  the  primary  of  the  solenoid  in  series  with 
six  volts  of  storage  battery.  The  secondary  winding 
should  be  connected  to  the  galvanometer. 

After  the  connections  have  been  made,  with  the  switches 
1,  2,  3  and  4  open  note  the  deflection  of  the  galvanometer 
when  the  circuit  is  closed  with  the  key  K.  Read  the  amme- 


20  MAGNETIC  INDUCTION  [55 

ter.     After  the  galvanometer  has  returned  to  zero,  again 
note  the  deflection  when  the  circuit  is  broken. 

Now  close  switch  number  1  and  note  the  deflection 
when  the  circuit  is  closed.  Repeat  the  ammeter  reading 
and  take  a  second  reading  of  the  deflection  when  the 
circuit  is  suddenly  opened  as  before.  Then  close  switch 
number  2,  leaving  number  1  closed,  and  repeat  the  scale 
and  ammeter  readings.  Continuing  in  this  way  secure 
scale  readings  for  five  different  values  of  the  current. 

(6)  Measure  the  inside  and  outside  diameter  of  the 
primary  winding  of  the  solenoid  with  the  calipers.  Note 
the  number  of  turns  n,  the  number  of  layers,  and  measure 
the  length  /. 


(c)  Using  your  data  and  formula  (1)  compute  H  for 
each  value  of  the  current.  Plot  a  curve  using  the  values 
of  the  galvanometer  deflections  as  abscissae  and  the  cor- 
responding changes  in  the  number  of  lines  of  force  (values 
of  HA)  as  ordinates. 

B  and  H  Curves. 

If  a  rod  of  some  magnetic  substance  longer  than  the 
solenoid  be  placed  within  the  coil,  the  magnetic  field 
intensity  B  within  the  rod  is  B  =  JJL  H  where  JJL  is  the 
permeability  of  the  substance.  The  total  number  of  lines 
<t>  threading  the  coil  is 

(2)     Ba  +  H  (A  -  a)  =  0 
where  a  is  the  cross  sectional  area  of  the  rod. 

The  value  of  A$  and  0,  ($  =  S  A  <£),  can  be  found  from 
the  plot  previously  made  and  the  scale  deflections  ob- 
served with  the  magnetic  substance  in  the  solenoid. 


55] 


MAGNETIZATION  OF  IRON 


21 


H  may  be  calculated  from  equation  (1)  as  before  and 
thus  B  may  be  calculated  from  equation  (2). 

To  get  a  series  of  values  of  B  and  H  and  at  the  same 
time  not  allow  them  to  be  reduced  to  zero  values  at  each 
step  it  will  be  necessary  to  proceed  by  increments;  i.  e. 
to  measure  the  deflections  produced  by  successive  changes 
in  current.  In  this  case  the  value  of  <j>  is  to  be  found 
by  summing  up  the  separate  increments,  A  0,  due  to 
successive  changes  in  current. 

The  following  table,  containing  partial  data  and  cal- 
culations, indicates  a  convenient  form  of  tabulating  the 
results. 


Reading     Deflection 

C  hange 
in  flux 

Total  flux 

I 

I 

7? 

Number 

(cms.) 

A,       </>  =  2A<£     ^P^S 

K,  closed 

+6.7 

+902.0 

+  902.0 

+0.15 

+  12.48 

+7950 

1 

+6.5 

+875.0 

+  1777.0 

+0.50 

+41.52 

+  15030 

2 

+2.6 

+350.0 

+2127.0 

+0.95 

+79.00 

+  17100 

3 

+  1.6 

+212.6 

+2339.6 

+  1.42 

+  118.00 

+  17860 

4 

+  1.3 

+  175.0 

+2514.6 

+  1.85 

+154.00 

+  18380 

4,  opened 

-1.4 

-  198.4 

+2316.2 

+  1.42 

+  118.00 

+  17650 

3 

-1.5 

-  202.0 

+2104.2 

+0.95 

+79.00 

+  16900 

2 

-2.1 

-  282.6 

+  1821.6 

+0.50 

+41.52 

+  15450 

1 

-3.7 

-  498.0 

+1323.6 

+0.15 

+  12.48 

+  11760 

K,      " 

-4.0 

-  538.5 

+795.1 

0.00 

+00.00 

+7300 

Reverse 

K,  closed 

-12.9 

-  1738.0 

-  942.9 

-0.15 

-  12.48 

-8260 

1 

-7.2 

-  970.0 

-  1912.9 

-0.50 

-41.52 

-  16280 

(a)  Measure  the  diameter  of  the  specimen  under  test 
with  the  calipers.     Demagnetize  it  by  slowly  withdrawing 
it  from  a  solenoid  operated  by  an  alternating  current,  and 
place  it  in  the  solenoid. 

(b)  Note   the   deflection   of   the   galvanometer    when 
the  circuit  is  closed,  and  read  the  ammeter.     Next,  leav- 
ing K  closed,  close  the  switch  number  1,  noting  the  deflec- 


22  MAGNETIZATION  OF  IRON  [55 

tion  as  before.  Read  the  ammeter.  Then  by  closing 
the  switches  in  the  order,  K,  1,  2,  3,  4,  and  opening 
them  in  the  reverse  order,  4,  3,  2,  1,  K,  reversing  the 
current  with  the  commutator  and  continuing  the  cycle 
take  a  series  of  readings  of  the  deflections  of  the  galvan- 
ometer and  corresponding  changes  of  the  current  such 
that  the  values  of  H  pass  from  zero  to  a  positive  maxi- 
mum back  to  zero  then  to  a  negative  maximum  and  back 
to  zero  then  again  to  a  positive  maximum.  Never  open 
any  switches  just  closed,  nor  close  any  just  opened,  in 
order  to  repeat  a  reading.  If  repetition  is  necessary  repeat 
the  entire  cycle,  including  the  demagnetization. 


(b)  From  the  calibration  curve  and  from  equation  (2) 
tabulate  and  plot  the  values  of  B  and  H  for  the  specimen 
furnished.  Tabulate  and  plot  the  values  of  JJL  and  H. 

Values  of  H  are  to  be  the  abscissae  in  all  the  curves. 


ELECTRICITY 

General  Directions. 

In  setting  up  any  electric  circuit  see  that  all  connections 
are  clean  and  well  made.  Never  make  connections  by 
twisting  ends  of  wire  together,  as  the  resistance  of  such  a 
connection  is  variable.  Always  use  the  binding  screws. 
Make  a  practice  of  arranging  wires  so  that  currents  op- 
posite directions  are  as  near  together  as  possible  in  order 
to  minimize  the  magnetic  effect. 

When  using  galvanometers  with  pivot  suspension  never 
move  or  leave  them  without  first  releasing  the  needle  from 
the  pivot. 

In  using  resistance  boxes,  take  care  to  protect  the  plugs 
and  the  tapered  holes  into  which  they  fit,  from  bruises 
and  dirt. 

The  brass  tapers  of  the  plugs  should  on  no  account  be 
touched  with  the  fingers  or  held  in  the  palm  of  the  hand; 
but  should  be  at  once  put  into  the  idle  sockets,  or,  if  these 
are  wanting,  the  plugs  may  be  stood  on  end  or  laid  on  any 
clean  surface. 

THEORY  OF  THE  TANGENT  GALVANOMETER. 

1.  A  unit  magnet  pole  is  defined  as  one  which,  when 
placed  in  air  at  a  distance  of  one  centimeter  from  a  pole 
of  equal  strength,  will  exert  upon  it  a  force  of  one  dyne. 

2.  A  unit  current  is  defined  (in  electro-magnetic  units) 
as  one  which,  flowing  in  a  conductor  one  centimeter  long, 
bent  into  an  arc  of  one  centimeter  radius,  will  act  on  a  unit 
magnet  pole  at  the  center  of  the  arc  with  a  force  of  one 
dyne. 


24  TEST  OF  THE  TANGENT  LAW  [61 

3.  We  shall  speak  of  the  field  at  the  center  of  the  coil, 
due  to  current  flowing  in  it,  as  the  force  in  dynes  exerted 
by  the  current  on  a  unit  magnet  pole  at  the  center  of  the 
coil :  let  this  be  denoted  by  F. 

4.  For  a  unit  current,   flowing  in  a  circular  coil  of 
1  turn,  radius  1  centimeter,  F=  27r;  for  n  turns,  F=  2-jrn. 

If  the  radius  of  the  coil  be  r,  F  =  27rn/r;  since  the  field 
due  to  a  given  length  of  wire  varies  as  1/r2,  but  the 
length  of  wire  in  the  coil  is  r  times  as  great.  This  value 
of  F,  namely  the  field  produced  at  the  center  of  the  coil 
by  a  unit  current  flowing  in  the  coil,  is  called  the  constant, 
G,  of  the  galvanometer. 

5.  Let    H  denote  the  horizontal   component  of   the 
earth's  magnetic  field  at  the  center  of  the  coil.     Then  if 
the  coil  be  set  with  its  plane  vertical  and  in  the  magnetic 
meridian,  the  two  fields  of  force  denoted  by  F  and  H  will 
act  at  right  angles  to  each  other,  upon  each  pole  of  the 
needle.    Assuming  the  needle  to  be  so  short  that  the  field 
at  each  pole  is  the  same  as  at  the  center  of  the  coil,  if  the 
current  I  cause  the  needle  to  deflect  through  an  angle  a 
from  its  zero  position,  it  may  be  shown  that  in  general 
tan  a  =  F/H  =IG/H.    From  this    I  =  (H  /  G)  tan  a. 

6.  I  is  here  expressed  in  c.  g.  s.  units.    The  commercial 
unit,  the  ampere,  is  0.1  of  the  c.  g.  s.  unit.    Hence  if  I  be 
expressed  in  amperes  I  =  (10  H  /  G  )  tan  a. 

10  H  I  G  is  called  the  reduction  factor,  K,  of  the  gal- 
vanometer. I  =  K  tana. 

61.     TEST  OF  THE  TANGENT  LAW. 

The  tangent  galvanometer  used  by  each  student  is  to 
be  set  with  its  needle  directly  over  a  particular  numbered 
brass  nail  in  the  table-top.  Record  the  number  of  the 
galvanometer  and  the  number  of  the  spot  assigned. 
Always  refer  to  the  galvanometer  by  number. 


61]  TEST  OF  THE  TANGENT  LAW  25 

(a)  Set  the  galvanometer  with  its  plane  in  the  magnetic 
meridian.  With  the  instrument  used  this  is  done  by  bring- 
ing the  ends  of  the  pointer  to  0°  and  180°. 

Connect  a  portable  storage  cell  in  series  with  the  gal- 
vanometer and  a  resistance  box,  and  insert  a  commutator 
in  the  circuit  for  reversing  the  direction  of  the  current  in 
the  galvanometer.  On  the  galvanometer  switch-board 
make  contact  with  the  50-turn  coil,  by  turning  in  the  screw 
marked  50,  the  other  screws  being  out  about  one  turn; 
the  current  then  flows  through  a  coil  of  50  turns. 

Take  out  plugs  from  the  resistance  box  until  the  deflec- 
tion is  about  50°.  Determine  the  deflection  by  reading 
both  ends  of  the  pointer  to  0.1°,  then  reversing  the  cur- 
rent to  get  a  deflection  in  the  opposite  direction  and 
reading  both  ends  again.  The  mean  of  the  four  readings 
is  the  deflection  required.  Increase  the  resistance  in  the 
box  so  as  to  obtain  deflections  at  intervals  of  5°  to  6° 
down  to  about  10°. 

In  what  follows,  the  electromotive  force  and  resistance 
of  the  cell  are  supposed  to  remain  constant  throughout 
the  experiment.  To  test  this,  repeat  the  first  reading. 
If  it  does  not  agree  with  the  former  value,  repeat  the  series 
in  the  same  order  as  before  to  a  point  where  the  new 
values  do  agree  with  the  old. 


(6)  Show  the  truth  of  the  relation  expressed  on  the 
preceding  page,  tan  a  =  F/  H.  By  combining  this  with 
Ohm's  law,  E  =  IR  where  R  is  the  total  resistance  in  the 
circuit,  show  that 

E/  K  =  R  tan  a  =  R/  cotan  a 

If  then  a  plot  be  made  with  resistances  as  abscissae  and 
cotangents  of  corresponding  angles  of  deflection  as  ordin- 

5 


26          REDUCTION  FACTOR  OF  GALVANOMETER     [61-62 

ates,  the  points  obtained  should  lie  on  a  straight  line  if 
the  galvanometer  obeys  the  tangent  law. 

(c)  Make  such  a  plot  from  your  observations,  using 
box  resistances  as  abscissae,  and  show  how  to  obtain  from 
it  the  constant  resistance  in  the  circuit  outside  the  box. 
Do  this  first  by  reading  it  directly  from  the  plot,  and  then 
more  accurately,  by  constructing  the  equation  of  the  line 
and  calculating  from  it. 

What  does  this  resistance  include? 

62.     REDUCTION  FACTOR  OF  GALVANOMETER. 
By  Deposition  of  Copper. 

(a)  Set  up  the  galvanometer  in  the  same  place  as 
before.  Record  the  number  of  the  galvanometer  and  the 
number  of  the  spot  assigned.  Straighten  out  two  of  the 
copper  wire  spirals  from  the  voltameter  cells,  clean  and 
smooth  them  well  with  fine  sand  paper,  and  rewind  on  a 
brass  tube  provided  for  the  purpose.  Do  not  touch  the 
wire  with  the  fingers  except  at  the  ends,  after  cleaning; 
catch  one  end  through  the  small  hole  in  the  brass  tube, 
and,  having  the  other  end  secured,  stretch  the  wire  while 
winding,  so  as  to  get  a  smooth  spiral.  If  the  surface  of 
the  wire  flakes  off  in  cleaning,  ask  for  new  wire.  Clean 
the  copper  plates  of  the  voltameters,  replace  them  and  the 
spirals,  and  fill  the  cell  with  fresh  electrolyte  solution. 

Connect  the  two  voltameters  in  series  with  a  portable 
storage  cell  and  the  5-turn  coil  of  the  galvanometer,  in 
such  a  way  that  copper  will  be  deposited  by  the  current 
on  the  spirals.  Insert  an  adjustable  rheostat  or  a  piece 
of  German  silver  wire  in  the  circuit  if  necessary  to  reduce 
the  deflection  to  about  25°.  A  current-reversing  switch 
should  also  be  placed  in  the  circuit,  so  that  reversing  the 


62]  REDUCTION  FACTOR  OF  GALVANOMETER          27 

direction  of  the  current  through  the  galvanometer  will  leave 
unchanged  the  direction  of  the  current  in  the  voltameters.  * 

(b)  After  the  circuit  has  been  closed  for  several  minutes 
lift  the  spirals  out  of  the  solution  and  see  if  they  are  uni- 
formly covered  with  a  bright  and  clean  deposit.  If  so 
plunge  them  at  once  into  a  beaker  of  clean  water,  then 
wash  thoroughly  under  the  tap  and  dry  by  gentle  heat,  no 
greater  than  may  be  easily  borne  by  the  hand. 

Weigh  the  spirals  separately  to  1  mg.  on  a  delicate 
balance.  Replace  the  spirals  in  the  voltameters,  put  them 
in  circuit  as  before  and  let  the  current  run  for  a  measured 
time  interval  not  less  than  twenty  minutes.  Keep  the 
deflection  constant  by  adjusting  the  resistance  in  the 
circuit,  reversing  the  direction  of  the  current  through  the 
galvanometer  at  about  the  middle  of  this  period.  Repeat 
the  washing,  drying  and  weighing.  The  gain  in  weight 
should  be  the  same  for  the  two  spirals.  If  it  is  nearly  the 
same,  take  the  mean. 


(c)  The  international  ampere,  the  unit  of  current,  is 
represented  sufficiently  well  for  practical  use  by  the  un- 
varying current  which,  when  passed  through  a  solution 
of  copper  sulphate  will  deposit  under  conditions  similar 
to  those  used  in  this  exercise,  0.000328  grams  of  copper 
per  second.    Using  this  fact,  calculate  the  reduction  factor 
for  the  coil  used. 

To  calculate  the  Constants    of   the    1-Turn,  5-Turn  and 
5Q-Turn  Coils,  From  Their  Measured  Dimensions. 

(d)  Measure  with  a  beam-compass  or  outside  calipers, 
the  diameter  of  the  ledge  on  the  face  of  the  ring,  which  is 

*If  available  a  direct  current  ammeter  may  be  placed  in  series 
in  this  circuit  and  its  calibration  tested  by  comparison  of  its  read- 
ing with  the  weight  of  copper  deposited. 


28  GALVANOMETER  CONSTANT?  [62 


the  same  as  the  diameter  of  the  bottom  of  the  groove,  or 
the  inner  diameter  of  the  coil.     Also  measure  the  outer  - 
diameter   of   the   wire   coil.      In   each   case   take   several 
measurements  along  different  diameters. 

The  first  layer,  in  the  bottom  of  the  groove,  consists  of  six  turns 
of  No.  14  wire,  so  connected  that  the  current  flows  through  all 
six  in  parallel,  thus  making  one  effective  turn.  This  is  the  1-turn 
coil. 

The  next  layer,  of  No.  16  wire,  consists  of  two  wires  in  parallel 
wound  four  times  around,  thus  making  four  turns.  These  eight 
wires  just  fill  the  width  of  the  groove,  as  do  the  six  wires  of  the 
first  layer.  The  four  turns  and  the  one  may  be  connected  in  series, 
making  the  5-turn  coil. 

Outside  of  this  are  ^45  turns  of  No.  22  wire,  wound  in  three  layers 
of  15  turns  each,  which  may  be  connected  in  series  with  the  five 
turns,  to  make  the  50-turn  coil. 

Samples  of  Nos.  14,  16  and  22  wire  are  furnished.  From 
the  measured  diameters  of  these  (insulated),  and  the  inner 
and  outer  diameters  of  the  whole  coil,  find  the  mean 
diameter  of  each  layer  of  wire. 

Calculate  the  constants  2w  n  /  r  for  the  1-,  4-,  and  each 
of  the  -three  layers  of  15-turns.  Giving  n  and  r  their 
appropriate  value  in  each  case. 

From  the  definitions  of  the  constant  it  is  evident  that 
the  constant  for  two  coils  in  series  is  the  sum  of  the  two 
constants  taken  separately.  Find  thus  the  required  con- 
stants for  the  5-turn,  and  50-turn  coils.  Also  calculate 
the  reduction  factors  of  the  1-turn  and  the  50-turn  coils. 

(e)  From  your  data  in  this  experiment  find  the  value 
of  H  at  the  point  where  the  needle  of  the  galvanometer 
is  located. 


63]  MEASUREMENT  OF  RESISTANCE  29 

63.     MEASUREMENT  OF  RESISTANCE. 
Wheatstone  Bridge. 

The  object  of  this  exercise  is  to  calibrate  a  resistance  box, 
using  a  slide-wire  Wheatstone  bridge  and  a  standard  resis- 
tance box ;  and  to  test  the  laws  for  resistances  connected  in 
series  and  in  parallel. 

(a)  Connect  a  I/eclanche  cell  to  the  bridge-wire  with 
a  key,  k,  in  the  circuit,  so  that  the  wire  forms  the  arms, 
p  and  q,  of  the  bridge.  To  eliminate  the  effect  of  induction 
and  capacity  in  any  parts  of  the  'apparatus  or  the  un- 
known resistance  the  key,  k,  in  the  battery  branch  should 

be  closed  before  the  key,  K, 
^XTV^  (usually  a  sliding  contact)  is 

pressed  and  should  be  kept 
closed  until  K  is  opened. 

To  avoid  polarization,  the 
battery  circuit  should  be 
closed  only  while  taking  an 
observation. 

.  2.-The  Wheatstone  Bridge.  Af ter  finding  the  resistance 

of  the  coils  separately,  deter- 
mine the  resistance  of  all  of  them  in  series  as  a  check. 

Interchange  the  known  and  unknown  resistances, 
leaving  the  connections  as  they  were,  and  repeat  the 
observations.  The  average  of  the  two  determinations  of 
the  length  p  and  the  length  q  (q  being  always  the  length 
opposite  the  unknown  resistance  x)  will  eliminate,  in 
part,  errors  due  to  inequalities  of  the  bridge  wire,  and 
the  resistance  of  the  connecting  wires. 

(6)  Determine  the  value  of  the  unknown  resistance 
furnished.  (Doorbell,  sounder,  relay,  or  incandescent 
lamp.)  Make  a  determination  of  the  resistance  of  two 
resistances  connected  in  parallel,  and  compare  your 
result  with  the  calculated  value. 


30 


COMPARISON   OF   RESISTANCE 


164 


64.     COMPARISON    OF    RESISTANCE 
Carey  Foster's  Method. 

The  following  extremely  accurate  method  of  comparing 
two  nearly  equal  resistances  was  devised  by  Foster. 

The  two  branches  p  and  q  of  the  Wheatstone  bridge 
are  extended  by  means  of  resistances  A  and  B,  using 
connecting  wires  of  resistance  a  and  b  respectively.  The 
resistances  r  and  x  should  be  of  about  the  same  magnitude 
as  A  or  B,  but  their  actual  value  need  not  be  known. 

Then  for  balance 
^_       (A+a+pk) 
x    ""    (B+b+qk). 

where  k  is  the  resistance 
per  centimeter  of  the 
wire.  If  A  and  B  are 
interchanged  together 
with  their  connecting 

Fig.  3. — Carey  Foster's  Method.  .  . 

wires,  and  a  new  posi- 
tion of  balance  found,  then,  r/x  =  (B+b+pfk)/(A+a+qfk). 

Adding  unity  to  each  member  of  the  equation  gives: 

r+x     A+a+pk  +  B+b+qk      B+b+p'k+A+a+q'k 
x  B+b+qk  A  +  a  +  q'k 

Remembering  that  p+q    =   p'+q',  we  see  that 

k(q  -  q')  =  (A  +  a)  -  (B  +  b)  =  A  -B+  (a-b).  Or  in  words 
the  resistance  of  the  part  of  the  bridge  wire  whose  length 
is  q  -  q'  is  equal  to  the  difference  between  the  resistances 
with  which  the  bridge  was  extended,  plus  the  difference 
in  the  resistances  of  the  connecting  wires. 

If  we  determine  two  corresponding  positions  of  balance 
#o   and   q'0,    with   no   resistances   in   the   extended   arms 


64]  ADDITIONAL  EXERCISES  31 

except   that   of  the  connecting   wires  later  to   be  used, 
k(q0—  q'o)  =  a—b,  and  by  substitution  we  have  finally, 

k[q-q'  ~  fao-g'o)]  =  A-B. 

This  is  the  principle  of  Carey  Foster's  method  of 
calibrating  a  bridge  wire,  and  with  a  calibrated  wire  is 
the  most  accurate  method  available  for  the  calibration 
of  resistances  by  comparison  with  standards. 

The  apparatus  is  provided  with  a  double-throw  com- 
mutator for  interchanging  A  and  B  together  with  their 
connecting  wires. 

(a)  To  determine  the  resistance  of  the  bridge  wire  by 
the  above  method  connect  a  standard  box  at  A  and  a  box 
which  is  later  to  be  calibrated  at  .5,  using  heavy  connecting 
wires  of  the  same  length.    Before  inserting  any  resistance 
from  either  box  in  the  circuit  (by  removing  plugs  or  ro- 
tating the  dial)  take  readings  both  direct  and  reversed; 
the  difference  in  the  two  settings  gives  a  length  of  wire 
having  a  resistance  equal  to  the  difference  in  resistance 
of  the  connections  as  they  now  stand.    This  length  should 
be  added,  algebraically,  to  the  following  readings  with  A. 

Make  A  =  0.1  ohm,  leaving  B  unchanged.  Find  a  rest 
point  for  both  the  direct  and  reversed  positions.  Increase 
the  value  of  A  by  0.1  ohm  steps  within  the  range  per- 
mitted by  the  resistance  of  the  bridge-wire. 

(b)  Compare  in  this  way  the  coils  in  the  box  B  with 
similar  coils  in  the  standard  box  A. 

ADDITIONAL    EXERCISES    WITH    THE    CAREY    FOSTER 

BRIDGE. 

1.     Determine  the  specific  resistance  of  some  pure  metal. 

After  checking  the  calibration  of  the  bridge-wire  in  (a) 
above,  calculate  the  specific  resistance  of  a  sample  of  wire 
of  measured  dimensions  from  the  readings  taken  with 


32      ABSOLUTE  DETERMINATION  OF  RESISTANCE  [64-65 

two  heavy  copper  staples  at  A  and  B,  and  the  readings 
with  a  measured  length  of  the  sample  inserted  in  series 
with  the  staple  A. 

2.  Determine  the  temperature  coefficient  of  resistance 
of  some  pure  metal. 

The  metal  is  furnished  in  the  form  of  a  coil  of  wire. 
Connect  the  coil  as  resistance  A  in  the  bridge  and  balance 
its  resistance  at  0°C.,  by  suitable  resistance  from  a  resis- 
tance box  as  B. 

Measure  the  resistance  at  100°C.,  or  some  measured 
higher  temperature  and  from  your  observations  compute 
the  value  of  the  temperature  coefficient  of  resistance  of 
the  metal. 

65.     ABSOLUTE  DETERMINATION  OF  RESISTANCE. 
Calorimeter  Method. 

Resistance  may  be  denned  as  that  property  of  a  con- 
ductor by  virtue  of  which  energy  is  expended  when  elec- 
tricity is  transferred  from  one  point  to  another  along 
the  conductor.  If  therefore  the  amount  of  energy  ex- 
pended can  be  measured,  the  resistance  may  be  deter- 
mined. Electrical  energy  may  be  transformed  into  energy 
of  motion,  chemical  separation,  or  heat.  If  the  conditions 
be  so  chosen  that  the  first  two  transformations  cannot 
occur  then  the  energy  will  all  be  transformed  into  heat  and 
can  readily  be  measured. 

A  unit  quantity  of  electricity  may  be  defined  as  the  quan- 
tity of  electricity  passing  a  given  point  in  a  conductor  in 
one  second  when  the  current  in  the  conductor  is  one 
unit.  Where  I  denotes  the  current  (assumed  to  be  con- 
stant) in  a  conductor,  Q,  the  quantity  passing  any  given 
point,  and  T  the  time,  then  we  have  the  equation, 
Q  =  I  T. 


65]      ABSOLUTE  DETERMINATION  OE  RESISTANCE     3S 

The  difference  of  potential  between  two  points  on  a 
conductor  is  defined  as  the  energy  which  would  be  ex^ 
pended  in  transferring  a  unit  quantity  of  electricity  from 
one  point  to  the  other.  Therefore  W  =  Q  (Vi  -  V,), 
where  W  is  the  energy  expended  and  Vi  -  V*  is  the,  dif- 
ference in  potential.  If  /  is  measured  in  amperes,  and 
Vi,  F2  in  volts,  then  W  is  measured  in  a  unit  called  the 
joule.  (1  joule  =  107  ergs).  We  have  then 

W     =  (Vi  -  V*)  I  T  x  107  ergs. 

From  Ohm's  law  I  R  =  Vi  -  V2,  R  being  the  resistance 
of  the  conductor  between  the  points  whose  difference  in 
potentials  is  Vi  -  V2. 

.'.  W  =  I2  R  T    x  107  ergs. 

Since  this  energy  is  all  expended  in  heating  the  con- 
ductor, W  =  H  J  where  H  is  the  amount  of  heat  pro- 
duced in  calories  and  /  is  the  mechanical  equivalent  of 

heat-  /.  H  J  =  I*  RT  x  107. 

Inserting  the  value  of  /,  (J  =  4.19  x  107),  and  solving 
for  R*  4.19  H 


(a)  To  determine  the  water  equivalent  of  the  thermos 
bottle  (which  is  to  be  used  as  a  calorimeter),  first  weigh 
it  with  thermometer  and  heating  coil  inserted.  Then 
add  about  25  gms.  of  hot  water,  replacing  the  coil  and 
thermometer,  and  determining  the  exact  amount  by 
weighing  the  bottle  after  the  water  has  been  added. 
Move  the  bottle  about  so  that  the  water  will  come  in 
contact  with  all  parts  of  the  inside,  hold  it  inverted 
the  greater  part  of  the  time  but  do  not  shake  it.  Note 
the  temperature  after  it  becomes  constant. 

Next  add  about  40-50  gms.  more  of  water  at  a  known 
temperature  near  that  of  the  room.  Determine  as 


34  ELECTROMOTIVE  FORCE  [65-66 

before  the  exact  amount  by  weighing  and  after  moving 
it  about  and  inverting  it  as  before  note  the  final  steady 
temperature.  Equating  the  heat  lost  by  the  bottle, 
thermometer,  coil,  and  water  first  put  in  to  that  gained 
by  the  water  added  later  will  give  the  desired  water- 
equivalent. 

(6)  Connect  the  heating  coil,  whose  resistance,  R, 
is  to  be  determined  in  series  with  an  ammeter,  leaving 
the  circuit  open.  Put  some  water  in  the  thermos  bottle, 
determining  the  exact  amount  by  weighing  the  bottle 
after  the  water  is  poured  in,  immerse  the  heating  coil 
and  thermometer  and  after  moving  the  bottle  about 
and  inverting  it  record  the  initial  temperature. 

Close  the  circuit  and  let  the  current  run  for  a  time 
interval,  T,  measured  in  seconds,  then  move  the  bottle 
about  to  secure  a  uniform  temperature  and  note  its 
final  constant  value.  , 

Make  a  second  trial,  using  your  results  as  a  guide  in 
varying  the  time  and  amount  of  water  used. 

(c)  Measure  the  resistance  of  the  heating  coil  using 
a  postoffice  bridge. 


(d)  From  your  data  in  (a)  and  (b)  calculate  the  resis- 
tance of  the  coil  used,  and  compare  your  results  with 
the  value  found  in  (c). 

66.     ELECTROMOTIVE  FORGE. 
Poggendorff's  Method.     The  Potentiometer. 

The  B.  M.  F.  of  a  cell  is  equal  to  the  maximum  differ- 
ence of  potential  which  it  is  capable  of  producing  at  its 
terminals.  This  maximum  occurs  when  the  external  cir- 
cuit is  broken,  i.  e.,  on  open  circuit. 


66]  ELECTROMOTIVE  FORCE  35 

Poggendorff's  method  of  measuring  the  E.  M.  F.  of  a 
cell,  b2,  consists  in  finding  two  points  A  and  C  in  an 
independent  circuit  (61,  G\t  A  B)  between  which  the  dif- 
ference in  potential  exactly  equals  that  of  the  terminals  of 
the  cell,  62,  on  open  circuit.  If  these  points  be  connected 
so  as  to  oppose  the  cell  no  current  will  flow  through  it 
and  the  difference  in  potential  between  the  two  points, 
calculated  from  the  current  and  resistance  between  A 
^-^  j  ^  and  C,  equals  the  B.  M.  F. 

~\§/ ^T        ~^\  of  the  cell. 

B        A   wire  or  other  resistance 


I 


like  the  wire  A  B  is 
called  a  potentiometer,  since 
for  constant  current  the  dif- 
ference in  potential  between 

Fig.  4.-Poggendorff's   method    of     twQ  pomts  js  proportional  tO 

measuring  electromotive-force.  v 

the  resistance  between  them. 

The  simplest  form  of  potentiometer  is  a  uniform  stretched 
wire.  If  the  points  of  balance  obtained  in  two  cases 
are  Cx  and  Cs,  while  the  corresponding  values  of  the 
E.  M.  F.  are  Ex  and  Es,  then 

_£*_        ACX 
Es          ACS' 

(a)  Connect  a  storage  cell  bl  in  series  with  a  tangent 
galvanometer,  Glt  and  a  high  resistance  wire,  AB,  whose 
resistance  per  centimeter  is  known.* 

Carefully  adjust  the  tangent  galvanometer.  Its  reduc- 
tion factor,  accurately  determined  for  its  present  position, 
is  given. 

Form  a  parallel  circuit  to  the  wire,  consisting  of  a  Le- 
clanche  cell,  62,  a  sensitive  galvanometer  G2,  and  a  key, 
at  C.  The  +  terminal  of  the  Leclanche  should  be  con- 


*If  the  resistance  of  the  wire  is  not  known  it  should  be  deter- 
mined, using  a  Postoffice  bridge. 


36  ELECTROMOTIVE  FORCE  [66 

nected  to  the  same  end  of  the  wire  as  the  +  terminal  of 
the  storage  battery. 

In  making  a  test,  set  the  slider  at  the  desired  point  on 
the  wire  and  press  the  key  just  long  enough  to  determine 
the  direction  of  the  deflection  produced,  as  a  long  con- 
tact will  cause  polarization  of  the  Leclanche. 

Determine  first  whether  the  Leclanche  is  properly  con- 
nected, which  will  be  the  case  if  it  is  possible  to  find  two 
points  of  contact  for  which  the  deflections  are  in  opposite 
directions.  Then  adjust  the  slider  until  the  galvanometer 
indicates  zero  current,  and  read  its  position.  Read  the 
tangent  galvanometer,  reversing  its  deflection  as  usual. 

As  the  error  in  your  result  will  be  directly  proportional 
to  the  error  in  the  determination  of  current  endeavor  to 
reduce  the  error  in  the  tangent  galvanometer  reading  to 
one  tenth  of  one  percent  or  less. 

From  Ohm's  law  calculate  the  difference  in  potential 
between  the  end  of  the  wire,  A,  and  the  sliding  contact,  C, 
which  is  equal  to  the  E.  M.  F.  of  the  Leclanche  cell. 

Replace  the  Leclanche  cell  with  a  dry  battery  and 
determine  its  E.  M.  F.  in  the  same  way. 

(6)  Check  your  determinations  of  the  E-  M.  F.  of  the 
cells  just  used  by  comparison  with  a  standard  cell  of 
known  E.  M.  F. 

Such  a  cell  is  used  as  a  standard  of  E.  M.  F.,  its  con- 
stancy depends  upon  careful  handling,  and  upon  having 
only  minute  currents  sent  through  it.  A  standard  cell  is 
intended  only  for  comparison  purposes  by  compensation 
or  open  circuit  methods.  Do  not  test  its  E.  M.  F.  on  a 
voltmeter. 

Knowing  that  the  E.  M.  F.  is  1.0183  volts,  it  will  be 
possible  by  a  little  calculation  to  set  the  slider  very 
near  to  the  true  point  at  the  start. 


66-67]  THERMO-ELECTROMOTIVE  FORCE  37 

Never  use  a  standard  cell  without  such  a  preliminary 
test  with  an  ordinary  cell.  If  it  is  impossible  to  approxi- 
mate closely  to  the  true  position  of  the  slider,  a  resistance 
of  several  thousand  ohms  should  be  included  in  series  with 
the  cell  to  protect  it  against  currents  of  dangerous  strength. 

67.     THERMO-ELEGTROMOTIVE  FORCE. 
Calibration  of  a  Galvanometer  as  a  Direct-Reading  Voltmeter 

Whenever  contact  is  established  between  two  dissimilar 
metals  an  electric  potential  difference  is  set  up  between 
their  "free"  ends.  If  now  these  free  ends  be  joined  this 
juncture  will  be  the  seat  of  a  new  difference  in  potential 
which  will  oppose  that  of  the  first  juncture.  The  magni- 
tude of  this  potential  difference  depends  not  only  upon  the 
metals  but  also  upon  the  temperature  of  the  juncture.  If 
therefore  the  two  junctures  be  maintained  at  different 
temperatures,  and  the  circuit  be  closed,  the  algebraic 
sum  of  the  electromotive  forces  in  the  circuit  will  not  be 
zero,  and  a  current  of  electricity  will  be  obtained.  The 
object  of  this  experiment  is  to  find  the  relation  between 
the  difference  in  temperature  of  the  junctures  and  the  di- 
rection and  magnitude  of  the  resulting  B.  M.  F.  for  a 
limited  range  of  temperature. 

(a)  Set  the  telescope  so  that  the  center  of  the  scale 
is  seen  in  the  galvanometer  mirror.  Connect  the  thermo- 
electric couples  in  series  with  the  galvanometer  and  an 
adjustable  resistance.  Keep  one  of  the  cups  at  the  tem- 
perature of  melting  ice  and  heat  the  other  until  the  water 
boils.  Adjust  the  resistance  until  the  galvanometer 
reading  is  on  the  scale  and  the  deflection  as  large  as 
possible. 

Then  remove  the  source  of  heat  and  take  the  galvano- 
meter readings  as  the  water  cools  for  about  every  10° 


38 


THERMO-ELECTROMOTIVE  FORCE 


[67 


change  in  temperature,  down  to  a  temperature  of  10°. 
Keep  the  water  in  the  cups  well  stirred  and  record  as  nearly 
simultaneously  as  possible  the  temperature  of  the  two 
junctures  and  the  corresponding  galvanometer  reading. 
Record  the  resistance  in  series  with  the  galvanometer  and 
also  the  distance  from  the  scale  to  the  galvanometer. 

(6)  In  order  to  find  the  potential  difference  in  volts 
between  the  terminals  of  the  thermo-electric  couples,  for 
any  given  temperatures,  it  will  be  necessary  to  calibrate 
the  galvanometer  as  a  voltmeter.  This  may  be  done  by 
finding  the  current  necessary  to  produce  deflections  over 
the  same  range  as  those  in  (a),  then  if  the  resistance  of 
the  circuit  is  known,  the  B.  M.  F.  may  be  found  from 
Ohm's  law. 

Pass  a  current  from  a  storage  battery,  B,  through  a 
resistance  box  containing  0.1,  0.2,  0.3,  0.4  ohm  coils  and 
an  ammeter,  A,  connected  in  series. 

Remove  the  thermo-couple  from  the  d' Arson  val 
galvanometer  circuit  and  attach  a  traveling  plug  to 
each  end  of  this  circuit.  Insert  these  traveling  plugs  on 
either  side  of  the  0.1  ohm  coil  whose  resistance  should 
be  in  the  circuit.  Leave  the  resistance  box,  R,  in  the 
d'Arsonval  galvanometer  circuit  with  the  resistance  un- 


Fig.  5. — Showing  the  connection  when  calibrating  the  d'Arsonval 
galvanometer  used  with  resistance  R  as  a  voltmeter. 


67]  THERMO-ELECTROMOTIVE  FORCE  39 

changed.  The  conditions  as  to  resistance  in  this  circuit 
must  be  the  same  as  in  (a).  The  resistance  of  the  thermo- 
couple is  small  as  compared  with  the  resistance  of  the 
remainder  of  the  circuit  and  may  be  neglected. 

The  resistance  of  the  d'Arsonval  galvanometer  circuit 
is  now  one  branch  of  a  parallel  circuit,  the  0.1  ohm 
resistance  of  the  box  between  the  galvanometer  terminals 
being  the  other  branch.  Since  the  resistance  in  the 
galvanometer  circuit  is  very  large  compared  to  this  0.1 
ohm,  the  difference  in  potential  between  the  terminals 
of  the  galvanometer  circuit  can  be  considered  as  the 
product  of  the  total  current  times  this  0.1  ohm  resistance. 
The  direction  of  the  current  should  be  such  as  to  give  a 
deflection  in  the  same  direction  as  the  deflection  in  (a). 

Adjust  the  resistance  in  the  box  connected  in  series 
with  the  storage  cell  to  give  a  suitable  deflection  of  the 
d'Arsonval  galvanometer.  Obtain  four  such  values 
covering  the  range  of  deflections  observed  in  (a),  reading 
the  ammeter  in  each  case. 

(c)  Knowing  the  direction  of  the  current  producing 
the  deflection,  find  which  way  the  current  was  flowing 
in  (a)  across  the  heated  juncture. 


(d)  Make  a  plot  of  the  results  in  (a)  plotting  deflections 
as  ordinates  and  differences  in  temperature  as  abscissae. 
From  the  data,  obtained  in  (b)  make  another  curve  on 
the  same  sheet  in  which  the  abscissae  give  the  electro- 
motive force  in  volts,  the  ordinates  being  scale  deflections 
as  before.  From  these  curves  find  the  E.  M.  F.  in  micro- 
volts for  a  single  element  when  the  difference  in  temperature 
between  the  junctures  is  1°C. 


40  THE  EARTH-INDUCTOR  [68 

68.     THE  EARTH-INDUCTOR. 
Relative  Calibration  of  a  Galvanometer. 

(a)  A  sensitive  galvanometer  is  necessary  in  this  ex- 
periment. Since  the  galvanometer  may  not  obey  any  sim- 
ple law,  such  as  the  tangent  law,  a  relative  calibration 
curve  must  be  obtained.  To  do  this  connect  a  portable 
storage  cell  in  series  with  a  stretched  wire  and  a  suitable 
resistance  coil;  set  the  galvanometer  in  the  position  in 
which  it  is  to  be  used  in  the  experiment.  If  the  galvano- 
meter terminals  be  now  connected  to  two  points  on  this 
stretched  wire,  separated  by  a  suitable  distance,  a  de- 
flection will  be  obtained.  Since  the  galvanometer  re- 
sistance is  high  compared  to  the  resistance  of  the  stretched 
wire  between  the  galvanometer  terminals,  it  follows  that 
the  current  through  the  galvanometer  is  approximately 
proportional  to  the  distance  between  the  galvanometer 
terminals.  Obtain  about  ten  galvanometer  deflections, 
reversing  the  current  through  the  galvanometer  in  each 
case,  note  also  the  distance  between  the  galvanometer 
terminals  along  the  stretched  wire  corresponding  to  each 
deflection.  Plot  your  results,  taking  relative  electromo- 
tive forces  as  abscissae  and  galvanometer  deflections  as 
ordinates. 

(6)  Connect  the  galvanometer  with  the  earth-inductor, 
placing  the  latter  as  far  as  possible  from  the  galvano- 
meter. Place  the  earth-inductor  so  that  the  two  sta- 
tionary, upright  supports  are  in  an  east-and-west  line  and 
set  the  circle  so  that  its  axis  of  rotation  is  horizontal. 
Turn  the  circle  slowly  into  a  horizontal  position,  let  the 
galvanometer  needle  come  to  rest,  and  then  turn  the  cir- 
cle suddenly  through  180°,  noting  the  effect  on  the  gal- 
vanometer. Explain  the  cause  of  the  current  produced. 


68]  THE  EARTH  INDUCTOR  41 

Turn  the  circle  in  the  same  direction  through  another 
180°,  and  compare  the  direction  of  the  induced  current 
with  that  just  obtained.  Account  for  the  directions  being 
as  you  find  them  in  the  two  cases. 

(c)  Rotate  the  coil  continuously  and  uniformly,   re- 
cording the  number  of  turns  per  minute  and  the  deflection 
of  the  galvanometer.     Now  rotate  the  coil  continuously 
and  uniformly  at  a  rate  either  one-half  or  twice  as  great 
as  the  rate  just  used,  recording  again  the  rate  and  the 
deflection.     From  the  calibration  curve  of  the  galvano- 
meter determine  the  relative  electromotive  forces  in  the 
two  cases.    Assuming  the  resistance  of  the  circuit  constant, 
what  effect  do  you  find  upon  the  induced  electromotive 
force  when  the  rate  of  rotation  is  doubled? 

When  the  coil  is  being  turned  at  a  constant  rate  are 
the  tubes  or  lines  of  magnetic  force  being  cut  at  a  constant 
rate?  Is  the  electromotive  force  induced  under  these 
conditions  constant  or  variable?  If  the  current  is  variable 
why  is  the  galvanometer  reading  fairly  constant  when 
the  turning  is  uniform? 

(d)  Set  the  coil  with  the  axis  of  rotation  vertical,  and 
rotate  the  coil  at  the  same  rate  as  in  one  of  the  cases  in 
(c).     To  what  component  of  the  earth's  magnetic  field  is 
the  induced  electromotive  force  proportional  in  this  case? 
To  what  component  was  it  proportional  in  (c)  ?    From  the 
relative  currents  obtained  in  the  two  cases  calculate  the 
angle  of  dip. 

(e)  By  varying  the  angle  of  inclination  of  the  coil,  find 
a  position  for  which  there  will  be  no  electromotive  force 
induced  when  the  coil  is  rotated.     Read  the  angle  of  in- 
clination of  the  axis  of  rotation  on  the  graduated  circle, 
and  compare  the  angle  thus  found  with  the  angle  of  dip 
calculated  in  (d). 


42  RESISTANCE  OF  AN  ELECTROLYTE  [68-69 

(/)  Turn  the  base  of  the  earth-inductor  through  90° 
and  rotate  the  coil  continuously  about  a  vertical  axis  at 
the  same  rate  as  in  (d).  Compare  the  deflection  here 
obtained  with  that  obtained  in  (d),  and  explain  the  dif- 
ference if  there  be  any  difference. 

(g)  From  the  absolute  value  of  H  find  the  total 
strength  of  the  earth's  field. 

69.     RESISTANCE  OF  AN  ELECTROLYTE. 
Kohlrausch's     Method.       Alternating     Current-Telephone. 

The  specific  resistance  of  a  material  is  the  resistance 
between  two  opposite  faces  of  a  cube  of  the  material  each 
edge  of  which  is  1  centimeter  in  length.  If  this  material 
is  a  substance  in  solution,  the  specific  resistance  always 
refers  to  a  particular  concentration  which  should  be 
stated.  The  reciprocal  of  the  specific  resistance  is  called 
the  specific  conductivity. 

Metallic  conductors,  as  far  as  we  know,  suffer  no  per- 
manent change  by  the  passage  of  electricity  through  them; 
in  electrolytes,  however,  the  transference  of  electricity 
from  the  anode  to  the  kathode  is  accompanied  by  the 
transport  of  matter. 

If  we  assume  that  the  electricity  is  carried  wholly  by 
the  positively  and  negatively  charged  ions  into  which 
some  of  the  molecules  are  split  up  when  in  solution  it 
follows  as  a  matter  of  course  that  the  ease  with  which 
the  current  will  flow  (as  measured  by  the  conductivity) 
in  a  solution  containing  a  definite  number  of  molecules, 
will  depend  on  the  number  which  are  dissociated. 

Let  us  take  as  the  /number  of  molecules,  the  number 
found  in  1  c.cm.  of  normal  solution  of  the  substance.  In 
a  0.5-normal  solution  this  number  would  be  scattered 


69]  RESISTANCE  OF  AN  ELECTROLYTE  43 

through  2  c.cm.  of  the  solution,  if  the  degree  of  dissocia- 
tion were  the  same,  the  specific  conductivity  would  be 
only  one-half  as  great,  since  only  half  as  many  ions  are 
present.  Dividing  the  specific  conductivity  by  the  con- 
centration will  give  us  the  conductivity  due  to  the  num- 
ber of  molecules  in  1  c.cm.  normal  solution — the  molecu- 
lar conductivity. 

The  limit  to  the  molecular  conductivity  will  evidently 
be  reached  when  all  the  molecules  are  dissociated. 

The  resistance  of  an  electrolyte  cannot  be  found  by  the 
ordinary  methods  since  polarization  enters,  producing  a 
back  E.  M.  F.  There  are  several  methods  for  measuring 
the  resistance  of  an  electrolyte,  one  of  these  methods,  due 
to  Kohlrausch,  almost  wholly  avoids  the  effects  due  to 
polarization  by  using  a  rapidly  alternating  current.  To 
secure  such  a  current  the  secondary  current  from  a  small 
induction  coil  will  be  used. 

(a)  Connect  a  storage  cell  to  the  primary  of  the  in- 
duction coil  and  connect  the  secondary  of  the  induction 
coil  to  the  bridge.  Place  the  electrolyte  in  one  branch 
and  a  resistance  box  in  the  other  branch.  Since  a  gal- 
vanometer will  not  detect  a  rapidly  alternating  current 
a  telephone  will  be  used  in  place  of  a  galvanometer.  Avoid 
any  loops  of  wire;  since  an  alternating  current  is  used, 
self-induction  is  present  and  should  be  made  as  small  as 
possible. 

If  k  be  the  specific  conductivity  of  a  solution  of  known 
concentration  and  W  the  resistance  of  this  electrolyte  be- 
tween the  fixed  electrodes  measured  with  the  bridge,  then 
the  constant,  h,  for  this  pair  of  electrodes  is  h  =  W  k. 

Measure  W,  using  a  1/50  normal  solution  of  KCL  The 
position  of  balance  will  be  the  position  in  which  there  is 


44  RESISTANCE  OF  AN  ELECTROLYTE  [69 

a  minimum  of  sound  in  the  telephone.    From  the  value  of 
k  given  in  the  tables  calculate  h. 

The  determination  of  k  for  the  various  concentrations 
of  the  substance  whose  conductivity  is  desired  involves 
only  measurement  of  W  for  each  concentration. 

(6)  Determine  the  resistance  of  the  substance  fur- 
nished for  the  following  concentrations:  3-,  2-,  1-,  0.1-, 
0.01-,  0.001-,  and  0.0001-  normal,  keeping  the  temper- 
ature constant. 

Determine  the  resistance  of  the  water  used  in  preparing 
the  solutions. 

Care  should  be  taken  not  to  disturb  the  electrodes 
while  changing  the  concentration  of  the  solution. 


(c)  Calculate  from  the  observed  values  of  the  resist- 
ance, the  specific  conductivity  and  the  molecular  conduc- 
tivity, presenting  the  results  in  tabular  form.  The 
conductivity  of  the  water  is  to  be  subtracted  from  that  of 
the  solutions. 


70]  POLARIZATION  EFFECTS  IN  A  CELL  45 

70.     STUDY  OF  POLARIZATION  EFFECTS  IN  A 

LECLANCHE  CELL. 
Galvanometer  Used  as  Voltmeter 

Use  a  d'Arsonval  galvanometer,  which  has  the  proper 
resistance  connected  in  series  with  it,  as  a  voltmeter.  In 
this  experiment  the  closing  of  a  key  produces  a  deflection 
of  the  galvanometer  corresponding  to  an  E.  M.  F.,  which 
is  changing  rather  rapidly,  so  that  a  reading  must  be 
obtained  as  soon  as  possible. 

(a)  Connect  the  terminals  of  the  Leclanche  cell  directly 
with  the  terminals  of  the  galvanometer,  and  so  get  a 
reading  which  is  proportional  to  the  total  E.  M.  F.,  E, 
of  the  cell. 

Throughout  the  succeeding  series  of  observations  the  volt- 
meter should  remain  connected  to  the  terminals  of  the  cell, 
except  that  it  should  be  disconnected  at  convenient  intervals 
to  check  the  zero  reading  on  the  scale. 

(6)  Connect  the  terminals  of  the  cell  through  a  spool 
of  4-ohms  resistance  and  a  break-circuit  key  (one  which 
breaks  connection  instead  of  making  it  on  depressing  the 
lever).  Complete  this  parallel  circuit  at  a  noted  time  and 
immediately  read  the  galvanometer  deflection,  e.  Be 
careful  not  to  connect  the  cell  in  such  a  way  as  to  send  a 
current  through  it  until  ready  to  begin  operations. 

(c)  At  the  end  of  five  minutes  read  e  again,  disconnect 
the  resistance,  by  means  of  the  break-circuit  key,  and  as 
quickly  as  possible  read  E.     This  reading  must  be  taken 
quickly  because  E,  which  has  been  reduced  by  polarization 
caused  by  the  passage  of  the  current,  immediately  begins 
to  return  to  its  former  value. 

(d)  Re-connect  the  resistance,  and  note  the  time  at 
which  e  again  has  the  same  value  as  that  read  in   (c). 
Count  from  this  another  5-minute  interval,  then  repeat 


46  COMPARISON  OF  CAPACITIES  [70-71 

the  readings  as  in  (c).  By  this  means  the  readings  ob- 
tained are  very  nearly  the  same  as  they  would  have 
been  if  no  time  had  been  lost  in  taking  them,  so  that  the 
current  could  flow  continuously. 

Continue  the  readings  in  this  way  until  four  periods  of 
five  minutes  each  have  been  covered. 

(e)  Leaving  the  resistance  disconnected,  after  taking 
the  last  reading  under  (d),  take  readings  of  E  at  5-minute 
intervals  for  about  45  minutes. 

At  the  end  of  this  interval  secure  a  cell  whose  E.  M.  F 
has  been  previously  determined,  or  is  known,  and  by 
getting  the  galvanometer  deflection  corresponding  to  its 
E.  M.  F.,  obtain  a  factor  for  reducing  your  readings  to 
volts. 

Why  could  not  a  standard  cell,  whose  resistance  is 
about  1,000  ohms,  be  used  for  this  purpose? 


(/)  Plot  a  curve  with  time  as  abscissae  and  corres- 
ponding values  of  E  as  ordinates,  for  the  whole  time  of 
the  experiment.  Remember  that  during  the  first  part,  the 
5-minute  intervals  only  are  to  be  counted. 

Derive  from  Ohm's  law  an  expression  for  the  resistance 
of  the  cell  in  terms  of  E,  e,  and  the  known  external  re- 
sistance. Using  this  expression  calculate  the  resistance 
of  the  cell  for  each  pair  of  values  of  E  and  e,  tabulate  the 
results,  and  show  that  for  this  purpose  the  scale  readings 
corresponding  to  E  and  e  may  be  used  without  knowing 
their  values  in  volts. 

71.     COMPARISON  OF  CAPACITIES. 

A  comparison  of  capacities  may  be  made  by  a  method 
which  is  analogous  to  the  Wheatstone  Bridge  method  of 
determining  the  resistance  of  electrolytes,  using  an  alter- 
nating current  and  telephone  (as  in  Experiment  69), 


71-72]  EFFICIENCY  OF  HEATING  DEVICES  47 

where  a  condition  of  balance  is  indicated  by  no  buzzing 
in  the  telephone. 

If  two  condensers,  whose  capacities  may  be  represented 
by  Ci  and  C2  be  placed  in  the  r  and  x  branches  of  the 
bridge  (see  fig.  2,  p.  29)  and  the  parts  of  the  slide  wire  be 
replaced  by  two  resistances  p  and  q  when  the  condition 
of  balance  is  obtained  it  may  be  shown  that  d/C2  =  q/p. 

(a)  Determine  the  capacity  of  the  condenser  fur- 
nished by  comparing  it  with  the  standard  condenser. 
The  resistance  box  p  should  have  a  range  of  from  1-10000 
ohms.  Give  to  p  a  fixed  value  and  vary  q  until  a  balance 
is  obtained.  Approach  the  position  of  balance  from  the 
side  of  too  large  resistances  as  well  as  from  the  side  of 
too  small  resistances,  using  the  mean  in  your  calculations. 

(6)  Change  the  dielectric  of  the  condenser  and  make  a 
second  determination  of  its  capacity. 


(c)  From  the  two  values  of  the  capacity  of  the  con- 
denser determine  the  specific  inductive  capacity  of  the 
substance  used  in  (b). 

(d)  Establish  the  truth  of  the  relation  used  in  calcula- 
ting the  capacity  of  the  condenser.     It  may  be  assumed, 
in  the  proof,  that  neither  absorption  nor  leakage  occur. 

72.     EFFICIENCY  OF   ELECTRICAL    HEATING    DEVICES. 

The  object  of  this  exercise  is  to  determine  the  efficiency  of 
a  hot  plate  when  used  to  heat  water. 

(a)  Connect  the  hot  plate  in  series  with  an  ammeter 
having  a  range  of  1  to  10  amperes  and  shunt  a  voltmeter 
having  a  range  of  1  to  120  volts  across  its  terminals. 

Weigh  the  kettle,   dry  and    empty,   and    again    when 


48  EFFICIENCY  OF  HEATING  DEVICES  [72 

about  half  full  of  water.  Close  the  circuit  through  the 
stove  and  let  the  current  run  for  one  minute  before  placing 
the  kettle  upon  it.  During  this  time  stir  the  water  in  the 
kettle  and  take  its  temperature. 

Start  the  stop  clock  as  the  kettle  is  placed  upon  the 
stove.  Read  the  voltmeter  and  the  ammeter  every 
minute,  arranging  the  readings  in  tabular  form  so  that 
the  average  may  be  conveniently  found  for  use  in  your 
computations.  Stir  the  water  occasionally. 

At  the  end  of  9i  minutes  begin  to  stir  the  water  and 
£  minute  later  read  the  temperature  of  the  water  in  the 
kettle  just  as  you  take  it  from  the  stove. 

Immediately  open  the  circuit. 

(b)  Make  a  second  determination  allowing  the  water 
to  come  to  the  boiling  point. 

(c)  Repeat   the   experiment   with   the   kettle   full   of 
water. 


(d)  From  your  average  ammeter  and  voltmeter  read- 
ings calculate  the  resistance  of  the  stove  and  using  this 
value  find  the  number  of  calories  consumed  from  the 
relation,  Heat  (in  calories)  =  0.24  C2  R  t  where  C  repre- 
sents the  current  in  amperes  and  R  the  resistance  in 
ohms  and  t  the  time  in  seconds.  Find  the  heat  received 
from  the  weight  of  water  and  the  temperature  change, 
taking  into  account  the  water  equivalent  of  the  kettle. 

Express  the  efficiency  of  the  stove  in  per  cent  in  each 
case.  At  local  rates  for  electricity  find  the  cost  of  one 
liter  of  boiling  hot  water. 


PHYSICAL  TABLES 


49 


DENSITIES    AND    THERMAL    PROPERTIES    OF    SOLIDS. 

(The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  substances  listed.  The  student  should  not  expect  the  prop- 
erties of  the  average  laboratory  specimen  to  correspond  exactly 
in  value  with  them.  As  a  rule  the  densities  are  given  for  ordi- 
nary atmospheric  temperature.  The  specific  heats  and  coefficients 
of  expansion  are  in  most  cases  the  average  values  between  0°  and 
100°C.  The  melting  points  and  heats  of  fusion  are  given  for 
atmospheric  pressure.) 

The  coefficient  of  cubical  expansion  of  solids  is  approximately 
three  times   the  linear  coefficient. 


CJ 

+J     .        £ 

OJ    bJO 

«+-4      ^ 

•a 

rt 

"3  S 

'O.g  8 

rt-S  d 

o  " 
•«->  -2 

Cj    w 

Solid. 

Q 

o,W 

CO 

o  °w 

I!*2 

o>  3 

cals.  per 

degrees 

cals.  per 

gms.  per  cc. 

gm. 

per  degree  C. 

C. 

gm. 

Acetamide 

1.56 

82 

Aluminum 

2.70 

0.219 

.0000231 

658 

Brass,  cast 

8.44 

.092 

.0000188 

"       drawn 

8.70 

.092 

.0000193 

Copper 

8.92 

.094 

.0000172 

1090 

43.0 

German-silver 

8.62 

.0946 

.000018 

860 

Glass,  common  tube 

2.46 

.186 

.0000086 

flint 

3.9 

.117 

.0000079 

Gold 

19.3 

.0316 

.0000144 

1065 

Hyposul.  of  soda 

1.73 

.445 

48 

Ice 

.918 

.502 

.000051 

0 

80. 

Iron,  cast 

7.4 

.113 

.0000106 

1100 

23-33 

"     wrought 

7.8 

.115 

.000012 

1600 

Lead 

11.3 

.0315 

.000029 

326 

5.4 

Mercury 

13.596 

.0319 

—39 

2.8 

Nickel 

8.90 

.109 

.0000128 

1480 

4.6 

Paraffin,  wax 

.90 

.560 

.000008-23 

52 

35.1 

liquid 

.710 

Platinum 

21.50 

.0324 

.0000090 

1760 

27.2 

Rubber,  hard 

1.22 

.331 

.000064 

Silver 

10.53 

.056 

.0000193 

960 

21.1 

Sodium  chloride 

2.17 

.214 

.000040 

800 

Steel 

7.8 

.118 

.000011 

1375 

Wood's  alloy,  solid 

9.78 

.0352 

75.5 

8.40 

"          "     ,  liquid 

.0426 

50 


PHYSICAL  TABLES 


SPECIFIC  RESISTANCE  OF  VARIOUS  SUBSTANCES. 


R=r(l+aO 

R= resistance  at  t°C. 

r=  resistance  at  0°C. 


Ohms  per 
mil  foot 
at  0°C. 


Specific 

Resistance 

Microhms  per 

centimeter 

cube 


Aluminum 17.38  2.889 

j  2400 

Carbon,  graphite I  42000 

Carbon,  arc  light 4000. 

Constantine _ 51. 

Copper,  annealed. 9.44  1.570 

Copper,  hard 9.64  1.603 

German  Silver 125.0  20.76 

Gold 2.07 

la.  la.,  hard 50.2 

la.  la.,  soft 47.1 

Iron 58.0  9.64 

Iron,  wire 58.0  9.50 

Iron,  telegraph  wire 90.0  15.0 

Lead  19.6 

Manganin 47.5 

Mercury 94.34 

Nickeline,  No.    I,  hard.  43.6 

Nickeline,  No.    I,  soft...  40.7 

Nickeline,  No.  II,  hard.  33.9 

Nickeline,  No.  II,  soft...  32.3 

Platinum  54.03  8.98 

Silver,  annealed 1.49 

Silver,  hard 1.62 

Steel 82.  13.3 

Liquids  at  18°C. 

Dilute  H  N  O3>  30%._ 129.  XlO4 

H2S04,  5% 486.  XlO4 

H2  S  04,  30%._ 137.  XlO4 

H2S  04,  80% 918.  XlO4 

Zn  S  04,  24 % 214.  XlO5 

Water 26.5X108 

Megohms 

Benzine 14.   X 106 

Ebonite „  28.   XlO9 

Glass,  20°C._ 91.   XlO6 

Glass,  200°C 22.7 

Gutta-percha,  24°C 4.5 XlO8 

Mica..._ 84.    XlO6 

Paraffine 34.   XlO9 

Paraffine,  oil._..  8.   XlO6 

Shellac 9.   XlO9 

Wood  Tar....  167.   XlO7 


Temperature 

Coefficient  a 

Divide  by  105 

309. 


388. 

28  to  44 
365. 
-1.1 
0.5 
453. 


387. 
±1. 
88. 
7.6 
7.7 
16.8 
18.1 
247. 
377. 


PHYSICAL  TABLES 


51 


(IN  Ohm-1  cm"1)  OF 


THIS   SPECIFIC   CONDUCTIVITY  k 

STANDARD-SOLUTIONS  FOR  DETERMINING 
RESISTANCE-CAPACITY  OF  VESSELS. 
(Kohlrausch  and  Holborn,  p.  204.) 


15° 

16° 
17° 
18° 
19° 
20° 
21° 
22° 
23° 
24° 
25° 
26° 
27° 
28° 
29° 
30° 


NaCl 

Saturated 
k 

0.2014 
0.2062 
0.2111 
0.2160 
0.2209 
0.2259 
0.2309 
0.2360 
0.2411 
0.2462 
0.2513 
0.2565 
0.2616 
0.2669 
0.2721 
0.2774 


KCl 

Normal 
k 

0.09252 
0.09441 
0.09631 
0.09822 
0.10014 
0.10207 
0.10400 
0.10594 
0.10789 
0.10984 
0.11180 
0.11377 
0.11574 


KCl 

KCl 

KCl 

0.1-Normal 

0.02-Normal 

0.01-Normal 

k 

k 

k 

0.01048 

0.002243 

0.001147 

0.01072 

0.002294 

0.001173 

0.01095 

0.002345 

0.001199 

0.01119 

0.002397 

0.001225 

0.01143 

0.002449 

0.001251 

0.01167 

0.002501 

0.001278 

0.01191 

0.002553 

0.001305 

0.01215 

0.002606 

0.001332 

0.01239 

0.002659 

0.001359 

0.01264 

0.002712 

0.001386 

0.01288 

0.002765 

0.001413 

0.01313 

0.002819 

0.001441 

0.01337 

0.002873 

0.001468 

0.01362 

0.002927 

0.001496 

0.01387 

0.002981 

0.001524 

0.01412 

0.003036 

0.001552 

Rate  of  Electrolytic  Deposition. 


Mg.  per 
Element.  Coulomb 

Aluminum 0935 

Copper 3282 

Gold.. 0679 

Hydrogen 0104 

Lead .1072 

Mercury.. 1 .038 


Mg.  per 
Element.  Coulomb 

Nickel 3043 

Oxygen 0831 

Platinum 1.0095 

Silver 1.1186 

Tin 6097 

Zinc....  .3365 


Specific  Inductive  Capacities. 


Medium 


Specific 
Inductive 
Capacity. 

Air,  760  mm .'. 1.0 

Alcohol 26.0 

Beeswax 1.8 

Ebonite 2.2-3.2 

Glass,  light  flint 6.72 

Glass,  dense  flint 7.38 

Glass,  hard  crown '   6.96 

Glass,  plate 5.8-8.5 

Gutta-percha 2.5 

Kerosene....  ..  2.-2.S 


Specific 

Medium  Inductive 

Capacity. 

Mica.. 6.64 

Paraffine,  solid 1.96-2.3 

Paraffine,  oil 1.92 

Petroleum 2.05 

Shellac. 2.7-3.7 

Sulphur 2.8-3.9 

Turpentine 2.2 

Vacuum 999 

Water....  76.0 


52 


NATURAL  SINES. 


0' 

6' 

12 

18 

24 

30' 

36 

42' 

48' 

54' 

123 

4   5 

0° 

0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12   15 

1 

2 
3 

0175 

0349 
0523 

0192 
0366 
0541 

0209 
0384 
0558 

0227 
0401 
0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
0663 

0332 
0506 
0680 

369 
369 
369 

12   15 
12   15 
12   15 

4 
5 
6 

0698 
0872 
1045 

0715 
0889 
1063 

0732 
0906 
1080 

0750 
0924 
1097 

0767 
0941 
i"5 

0785 
0958 
1132 

0802 
0976 
"49 

O8ig 

0993 
1  1(>7 

08-37 
ion 
1184 

0854 
1028 

I2OI 

369 
369 
369 

12   15 
12   14 
12   14 

7 
8 
9 

1219 

1392 
1564 

1236 
1409 
1582 

1754 

1253 
1426 

'599 

1271 
1444 
1616 

1288 
1461 
1633 

1305 
1478 
1650 

*323 
1495 
1668 

1340 
1513 
I685 

1357 
1530 
1702 

1374 
1547 
1719 

369 
369 
369 

12   14 
12   14 
12   14 

10 

1736 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

369 

12   14 

11 
12 
13 

1908 

2079 
2250 

1925 
2096 
2267 

1942 
2113 
2284 

T959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

2OII 

2181 

2351 

2028 
2198 
2368 

2045 
2215 
2385 

2062 
2232 
24O2 

369 
369 
368 

II   14 
II   14 
II   14 

14 
15 
16 

24*19 
2588 
2756 

2436 
2605 
2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 
2857 

2538 
2706 
2874 

2554 
2723 
2890 

2571 
2740 
2907 

368 

368 
368 

II   14 
II   14 
II   14 

17 
18 
19 

2924 
3090 
3256 

2940 
3107 
3272 

2957 
3123 
3289 

2974 
3140 
3305 

2990 

3156 
3322 

3007 
3173 
3338 

3024 
3T90 

3355 

3040 
3206 
3371 

3057 
3223 

3387 

3074 
3239 
3404 

368 

368 

3  5  8 

II   14 
II   14 
II   14 

20 

3420 

3437 

3453 

3469 

3486 

3502 

35i8 

3535 

355i 

3567 

3  5  8 

II   14 

21 
22 
23 

35B4 
3746 
39°7 

3600 
3762 
3923 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 

3971 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

3714 
3875 
4035 

3730 
3891 
4051 

3  5  8 
3  5  8 
3  5  8 

II   14 
II   14 
II   14 

24 
25 
26 

4067 
4226 
4384 

4083 
4242 
4399 

4099 
4258 
4415 

4"5 
4274 
4431 

4i3i 
4289 
4446 

4M7 
4305 
4462 

4163 
4321 

4478 

4179 
4337 
4493 

4195 
4352 
4509 

42IO 

4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II   13 
II   13 

10  13 

27 
28 

29 

4540 
4695 

4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
4741 
4894 

4602 
4756 
4909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
4970 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 
10  13 

30 
~31~ 
32 
33 

5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

3  5  8 

10  13 

5150 
5299 
5446 

5592 
5736 
5878 

5165 
53M 
546i 

5180 
5329 
5476 

5195 
5344 
5490 

5210 

5358 
5505 

5225 
5373 
5519 

5240 

5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5707 
5850 
5990 

5284 
5432 

5577 

257 
2  5  7 

2  5  7 

IO   12 
10   12 
10   12 

34 
35 
36 

5606 
5750 
5892 

5621 

5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

572i 
5864 
6004 

257 

2  5  7 
2  5  7 

10   12 
IO   12 

9  12 

37 
38 
39 

6018 

6i57 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 
6347 

6088 
6225 
6361 

6101 
6239 
6374 

6115 
6252 

6388 

6129 
6266 
6401 

6143 
6280 
6414 

2  5  7 
257 
247 

9  12 
9  ii 
9  ii 

40 

£428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

247 

9  ii 

41 

42 
43 

6561 
6691 
6820 

6574 
6704 

6833 

6587 
6717 

6845 

6600 
6730 
6858 

6613 
6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

6652 
6782 
6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
2  4  6 
2  4  6 

9  ii 
9  ii 
8  ii 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

2  4  6 

8  10 

NATURAL  SINES. 


53 


0' 

6' 

12' 

18 

24 

30 

36 

42' 

48 

54 

123 

4      5 

45° 

7071 

7083 

7096 

7108 

7120 

7133 

7M5 

7157 

7169 

7181 

2  4  6 

8     10 

46 
47 
48 

7193 
73M 
743i 

7206 

7325 
7443 

7218 
7337 
7455 

7230 
7349 
7466 

7242 
7361 

7478 

7254 
7373 
7490 

7266 
7385 
750i 

7278 
7396 
7513 

7290 
7408 
7524 

7302 
7420 
7536 

2  4  6 
246 
246 

8     10 
8     10 
8     10 

49 
50 
51 
52 
53 
54 

7547 
7660 
777i 

7558 
7672 
7782 

7570 
7683 
1793 
7902 
8007 
8in 

758i 
7694 
7804 

7593 
7705 
7815 

7923 
8028 
8131 

7604 
7716 
7826 

7615 

7727 
7837 

7627 
7738 
7848 

7638 
7749 
7859 

7649 
7760 
7869 

246 
246 
245 

8      9 
7      9 
7      9 

7880 
7986 
8090 

7891 
7997 
8100 

7912 
8018 
8121 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

245 
235 
235 

7      9 
7      9 

7      8 

55 

8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

235 

7      8 

56 
57 
58 

8290 

838? 
8480 

8572 
8660 

8746 

5829 
8910 

8988 

8300 
8396 
8490 

8310 
8406 
8499 

8320 
8415 
8508 

8329 
8425 
8517 
8607 
8695 
8780 

8862 
8942 
9018 

8339 
8434 
8526 

8616 
8704 
8788 
8870 
8949 
9026 

834« 
8443 
8536 

8358 
8453 
8545 

8368 
8462 
8554 

8377 
8471 
8563 

2  3  5 
235 
235 

6       8 
6      8 
6      8 

59 
60 
61 

S58i 
8669 
8755 
8838 
8918 
8996 

8590 
8678 
8763 

8599 
8686 

8771 

8854 

8934 
9011 

8625 
8712 
8796 

8634 
8721 
8805 

8643 
8729 
8813 

8652 
8738 
8821 

i  3  4 
i  3  4 

i  3  4 

6      7 
6      7 
6      7 

62 

63 
64 

8846 
8926 
9<>Q3 
9078 

8878 
8957 
9033 

8886 
8965 
9041 

8894 
8973 
9048 

8902 
8980 
9056 

i  3  4 
i  3  4 
i  3  4 

5      7 
5      6 
5       6 

65 

9063 

9070 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

I  2  4 

5       6 

66 
67 
68 

9135 

9205 

9272 

9M3 
9212 
9278 

9150 
9219 
9285 

9157 
9225 
9291 

9164 
9232 
9298 

9171 
9239 
9304 

9178 
9245 
93ii 

9184 
9252 
9317 

9191 
9259 
9323 

919^ 
9265 
9330 

i  2  3 
i  2  3 
i  2  3 

5       6 
4      6 
4       5 

69 
70 
71 

9336 

9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
9415 
9472 

9361 
9421 

9478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 

9391 
9449 
9505 

i  2  3 
i  2  3 
i  2  3 

4      5 
4       5 
4       5 

9500 

72 
73 
74 

95ii 
9563 
9613 

9516 
9568 
9617 

952i 
9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9548 
9598 
9646 

9553 
9603 
9650 

9558 
9608 
9655 

i  2  3 

122 
122 

4      4 
3      4 
3      4 

75 

9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

112 

3      4 

76 
77 
78 

9703 
9744 
9781 

9707 
9748 
9785 

9711 
9751 
9789 

9715 
9755 
9792 

9720 

9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9736 
9774 
9810 

9740 
9778 
9813 

I  2 
I  2 
I  2 

3      3 
3      3 

2       3 

79 
80 
81 

9816 
9848 
9877 

9820 

9851 
9880 

9823 

9854 
9882 

9826 

9857 
9885 

9829 
;S6o 
9888 
9912 
9934 
9952 

9833 
9863 
9890 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I  2 
O  I 

o  i 

2        3 
2         2 
2         2 

82 
83 
84 

9903 
9925 
9945 

9905 
9928 

9947 

9907 
9930 
9949 
9965 

9910 
9932 
9951 
9966 

9914 
9936 
9954 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

0  I 
0  I 
P  I 

2         2 

2 

I 

85 

9962 

9963 

9968 

9969 

9971 

9972 

9973 

9974 

O  O  I 

I 

86 
87 
88 

89 

9976 
9986 
9994 

9977 
9987 
9995 

9978 
9988 
9995 

9999 

9979 
9989 
9996 

9980 
9990 
9996 

9981 
9990 
9997 

1.000 
nearly 

9982 
9991 
9997 

9983 
9992 

9997 

9984 

9993 
9998 

9985 
9993 
9998 

0  0  I 

o  o  o 
o  o  o 

I 
I 
O        O 

9998 

9999 

9999 

9999 

1.  000 

nearly 

1.000 

nearl" 

I.OOO 
nearly 

I.OOO 
nearly 

000 

0        0 

54 


NATURAL  TANGENTS. 


0' 

6' 

12 

18 

24' 

30' 

36 

42' 

48' 

54' 

123 

4      5 

0° 

.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12         I4 

1 
2 
3 

.0175 

•0349 
.0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
0577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

0314 
0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12         I5 

1         "5 
i         15 

4 
5 
6 

.0699 
.0875 
.1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
"39 

0805 
0981 
"57 

0822 
0998 
"75 

0840 
1016 
1192 

0857 
1033 

1210 

369 
369 
369 

i         15 
i         IS 

i         i5 

7 
8 
9 

.1228 
.1405 
.1584 

1246 

1423 
1602 

1263 
1441 
1620 

1281 

1459 
1638 

1299 
1477 
1655 

1317 
1495 
1673 

1334 
1512 
1691 

'352 
1530 
1709 

1370 
1548 
1727 

1388 
1566 
1745 

369 
369 
369 

i         15 
i         '5 
i         15 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

369 

12         15 

11 
12 
13 

.1944 

.2126 
.2309 

1962 
2144 
2327 

1980 
2162 
2345 

1998 
2180 
2364 

2016 
2199 

2382 

2035 

2217 

2401 

2053 
2235 
2419 

2071 
2254 
2438 

2089 
2272 
2456 

2107 
2290 

247_5 
2661 
2849 
3038 
3230 
3424 
3620 

369 
369 
369 

12         15 
12         15 
12         15 

14 
15 
16 

•2493 
.2679 
.2867 

2512 
2698 
2886 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 
2773 

2962 

2605 
2792 
2981 

2623 
28ll 
3000 

2642 
2830 
3019 
3211 

3404 
3600 

3     6     9 
369 
369 

12         l6 

13       16 
13       16 

17 
18 
19 

•3057 
3249 

•3443 

3076 
3269 
3463 

3096 
3288 
3482 

3"5 
3307 
3502 

3134 
3327 
3522 

3153 
3346 
3541 

3172 
3365 
356i 

3191 

3385 
3581 

3     6    10 

3     6    10 
3     6    10 

13      16 
13       16 
13      17 

20 

•3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3     7    1° 

>3       »7 

21 
22 
23 

•3839 
.4040 

•4245 

3859 
4061 
4265 

3879 
4081 

4286 

3899 
4101 
4307 

39'9 
4122 

4327 

3939 
4142 

4348 

3959 
4163 
4369 

3978 
4183 
4390 

4000 
4204 
44" 

4020 
4224 
4431 

3     7    10 
3     7    10 
3     7    10 

13       '7 
>4       '7 
14       17 

24 
25 
26 

•4452 
.4663 
.4877 

4473 
4684 
4899 

4494 
4706 
4921 

4515 
4727 
4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 
4791 
5008 

4599 
4813 
5029 

4621 

4834 
5051 

4642 
4856 
5073 

4      7    10 
4      7    ii 
4      7    ii 

14       18 
14      18 
15      18 

27 
28 
29 

•5095 
•5317 

•5543 

5H7 
3340 
5566 

5139 
5362 
5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

5250 
5475 
5704 

5272 
5498 

5727 

5295 
5520 
5750 

4      7    H 
8    ii 

8      12 

15      18 
15      >9 
'5       '9 

30 

•5774 

5797 

5820 

5844 

5867 

5890 

59M 

5938 

596i 

5985 

8      12 

16       20 

31 
32 
33 

.6009 
.6249 
.6494 

6032 
6273 
6519 

6056 
6297 
6544 

6080 
6322 
6569 

6104 
6346 
6594 

6128 

6371 
6619 

6152 

6395 
6644 

6176 
6420 
6669 
6924 
7186 
7454 

6200 

6445 
6694 

6224 
6469 
6720 
6976 

7239 
7508 

8      12 
8      12 

S    13 

16         20 
10         20 
17         21 

34 
35 
36 

6745 
.7002 
.7265 

6771 
7028 
7292 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 
7373 

6873 
7133 
7400 

6899 
7159 
7427 

6950 
7212 
7481 

9    '3 

4      9    »3 
5      9    »4 

17         21 

l8         22 
l8         23 

37 
38 
39 

•7536 
•7813 
.8098 

7563 
7841 
8127 

7590 
7869 
8156 

7618 
7898 
8185 

7646 
7926 

8214 

7673 
7954 

8243 

7701 
7983 
8273 

7729 
8012 
8302 

7757 
8040 
8332 

7785 
8069 
8361 

5      9    J4 
5      °    '4 
5      o    15 

18      23 
19      24 

20       =4 

40 

.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

5      o    15 

20       25 

41 
42 
43 

.8693 
.9004 
•9325 

8724 
9036 
9358 

8754 
9067 
9391 

8785 
9099 
9424 

8816 
9131 

9457 

8847 
9163 
9490 

8878 
9195 
9523 

8910 
9228 
9556 

8941 
9260 
9590 

8972 
9293 
9623 

9965 

5     o    16 
5      i    16 
6      i    17 

21          26 
ZI          27 
22         28 

44 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

6    n    17 

23          29 

NATURAL  TANGENTS. 


55 


45° 

46 
47 
48 

0 

6' 

12 

18 

24' 

30 

36 

42' 

48 

54' 

123 

4  5 

1.  0000 

0035 

0070 

0103 

0141 

0176 

0212 

0247 

0283 

0319 

6    12    18 

.24  3° 

1-0355 
1.0724 
1.1106 

0392 
0761 
H45 

0428 
0799 
1184 

0464 

0837 
1224 

0501 

0875 
1263 

0538 
0913 
1303 

0575 
0951 
1343 

0612 
0990 
1383 

0649 
1028 
1423 

0686 
1067 
1463 

6    12    18 
6   43    19 
7    13   20 

25  3' 
25  3« 
26  33 

49 
50 
51 
52 
53 
54 
~55 

1.1504 

1.1918 

1-2349 

1544 
1960 
2393 

1585 

2OO2 

2437 

1626 
2045 

2482 

1667 
2088 
2527 

1708 
2131 
2572 

I75C 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 
2753 

7    '4    21 
7    14    2* 
8    15    23 

28  34 
29  36 
30  38 

1.2799 
1.3270 
1-3764 

2846 
3319 
3814 

2892 
3367 
3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
3514 
4019 

3079 
3564 
4071 

3127 
36i3 
4124 

3175 
3663 
4176 

3222 

3713 
4229 

8    16    23 
8    16    25 
9    17    26 

3»  39 
33  4> 

34  43 

1.4281 

4335 

4388 
4938 
5517 
6128 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9    18   27 

36  45 

56 
57 
58 

1.4826 

1-5399 
1.6003 

4882 
5458 
6066 

4994 
5577 
6191 

5051 
5637 
6255 

5108 

5697 
6319 

5166 

5757 
6383 

5224 
5818 
6447 

5282 
5880 
6512 

5340 
594i 
6577 

ip    19   29 

10     20     30 
II      21      32 

38  48 
40  50 
43  53 

59 
60 
61 
62 
63 
64 
65 

1.6643 
1.7321 
1.8040 

6709 

7391 
8115 

6775 
7461 
8190 

6842 
7532 
8265 

6909 
7603 
8341 

6977 

7675 
8418 

7045 
7747 
8495 

7"3 
7820 

8572 

7182 

7893 
8650 

725' 
7966 
8728 

11    23    34 

12     24     36 

13  26  38 

45  56 
48  60 
Si  64 

1.8807 
1.9626 
2.0503 

8887 
9711 
0594 

8967 

9797 
0686 

9047 
9883 
0778 

9128 
9970 
0872 

9210 
0057 
0965 

5292 
0145 
1060 

9375 
6233 

"55 

2148 

9458 
0323 
1251 

9542 
0413 
1348 

14  27  41 

»5    29    44 

16    31    47 

55  68 
58  73 
63  78 

2.1445 

1543 

1642 

1742 

1842 

1943 

2045 

2251 

2355 

i?    34    5» 

68  85 

66 
67 
68 

2.2460 
2-3559 
2-4751 

2566 

3673 
4876 

2673 

3789 
5002 

2781 
3906 
5129 

2889 
4023 
5257 

2998 
4142 
5386 

3IP9 
4262 

5517 

3220 
4383 
5649 

3332 
4504 
5782 

3445 
4627 
59l6 

18    37    55 
20   40   60 
22    43    65 

74  92 
79  99 

87  108 

69 
70 
71 
72 
73 
74 
75 

2.6051 

2-7475 
2.9042 

6187 
7625 
9208 

6325 
7776 
9375 

6464 
7929 
9544 

6605 
8083 
9714 

6746 
8239 
9887 

6889 
8397 
6061 

7034 
8556 
0237 

7179 
8716 
0415 

7326 
8878 
0595 

24    47    7' 
26    52    78 

29    58    87 

95  "8 
104  130 
"5  M4 

3-0777 
32709 
34874 

0961 
2914 
5105 

1146 
3122 
5339 

1334 
3332 
5576 

1524 

3544 
5816 

1716 
3759 
6059 

1910 
3977 
6305 

2106 
4197 
6554 

2305 
4420 
6806 

2506 
4646 
7062 

32    64   96 
36   72  108 

41     82  122 

129  1161 
144  180 
162  203 

3-7321 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

9812 

46    94  139 

i  86  232 

76 
77 
78 

4.0108 

4-33I5 
4.7046 

0408 
3662 
7453 

0713 
4015 
7867 

J022 

4374 
8288 

1335 
4737 
8716 

1653 
5107 
9152 

1976 
5483 
9594 

2303 
5864 
0045 

2635 
6252 
0504 

2972 
6646 
0970 

53  107  160 
62  124  186 
73  146  219 

214  267 
248  310 
292  365 

79 
80 
81 

5.1446 
5-6713 
6.3i3fc 

1929 
7297 
3859 

2422 
7894 
4596 

2924 
8502 
5350 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
0405 
7920 

5026 
1066 

8548 

557« 
1742 
9395 

6140 
?13?, 

87  175  262 

350  437 

6264 

Difference  -  col- 
umns  cease  to  be 
useful,   owing    to 
the    rapidity  with 
which    the    value 
of  the  tangent 
changes. 

82 
83 
84 

7-II54 
8.1443 
95M4 

2066 
2636 
9.677 

3002 
3863 
9-845 

3962 
5126 
IO.O2 

4947 
6427 

10.20 

5958 
7769 
10.39 

6996 
9152 
10.58 

8062 

0579 
10.78 

9158 
2052 
10.99 

0285 
3572 

11.20 

85 
86 
87 
88 

89 

U-43 

ri.66 

11.91 

I2.I6 

1243 

12.71 

13.00 

13-30 

13.62 

13-95 

14.30 
19.08 
28.64 

14.67 
19.74 
30.14 

15-06 
20.45 
31.82 

15.46 
21.20 
33-69 

15.89 

22.02 
35-80 

16.35 
22.90 
38.19 

16.83 
23.86 
40.92 

17-34 
24.90 
44.07 

17.89 
26.03 
47-74 

18.46 
27.27 
52.08 

57-29 

63.66 

71.62 

81.85 

95-49 

114.6 

143.2 

[91.0 

286,5 

573-c 

56 


LOGARITHMS. 


0 

1 

2 

3 

4 

•5 

6 

7 

8 

9 

123 

456 

789 

10 

oooc 

0043 

oose 

OI2 

0170 

0212 

025 

029^ 

0334 

0374 

Use  Table  on  p.  58 

11 

12 
13 
14 
15 

16 

041 
079 

"3 

0453 
0828 

"73 

0492 
0864 
1206 

053 
089 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

064 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 
7430 

4811 
3710 
3610 

15  19  23 

14  17  21 

13  16  19 

26  30  34 
24  28  31 
23  26  29 

146 
176 
204 

1492 
1790 
2068 

1523 

1818 
209^ 

155 
184 
2122 

i5«4 
'875 
2148 

1614 
1903 
2175 

1644 
193 

220 

1673 

1959 
2227 

17^3 
1967 

2253 

1732 
2014 
2279 

36  9 
36  8 
3  5  8 

12  15  18 
II  14  17 

21  24  27 
2O  22  25 

18  21  24 

17 
18 
19 

2304 

2553 
2788 

2330 

2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 

2878 

2430 
2672 
2900 

2455 
2695 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

257 
257 
247 

IO  12  15 

9  12  14 
9  "  13 

17  20  22 

16  19  21 
16  18  20 

20 

3OI9 

3032 

3054 

3075 

3096 

3H8 

3139 

3160 

3181 

320 

246 

8  ii  13 

15  17  19 

21 
22 
23 

3222 

3424 
3617 

3243 
3444 
3636 

3263 
3464 

3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
356o 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

2  4  6 
246 
246 

8  10  12 
8  10  12 

7  9  n 

14  16  18 
14  15  i-7 
13  '5  17 

24 
25 
26 

3802 
3979 
4i$6 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

39<>9 
4082 

4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

245 
235 
235 

7  9  ii 
7  9  10 
7  8  10 

12  14  16 
12  14  15 
ii  13  15 

27 
28 
29 

4314 
4472 
4624 

4330 
4487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

235 
235 
*  3  4 

689 
689 
679 

ii  13  14 

II  12  I. 
10  12  13 

30 
31 

33 

477i 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1  3  4 

6  7  9 

10  ii  13 

4914 
051 

185 

J928 
5065 
198 

4942 

5079 
5211 

-f955 
5092 
5224 

4969 
105 
237 

4983 
5"9 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

1  3  4 
134 
*  3  4 

6  7  8 

5  7  8 
5  6  8 

10  II  12 

9  ii  12 
9  10  12 

34 
35 
36 
37 
38 
39 

315 
441 

563 

328 
453 
575 

5340 
5465 
5587 

5353 
5478 
5599 

366 
490 
611 

5378 
5502 
5623 

5391 

5514 
5635 

5403 
5527 
5647 

54i6 

5539 
5658 

5428 
555i 
5670 

i  3  4 
i       4 
i       4 

5  6  8 
5  6  7 
5  6  7 

9  10  ii 
9  jo  ii 
8  10  ii 

682 
798 
911 

694 
SSog 
922 

5705 
5821 

5933 

5717 
5832 
5944 

729 
843 
955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 

5999 

5786 

5899 
6010 

i       3 
1       3 

i       3 

567 
5  6  7 
457 

8  9  10 
8  910 
8  910 

40 

021 

o3i 

6042 

605.3 

6064 

6075 

6085 

6096 

6107 

6117 

1       3 

4  5  6 

8  9  10 

41 

42 
43 

128 

232 

335 

138 
243 
345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
274 
375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 

6314 
6415 

6222 

6325 
6425 

i       3 
i       3 
r       3 

4  5  6 
4  5  6 
4  5  6 

789 

7  8  9 
7  8  g 

44 
45 
46 
47 
48 
49 

435 
532 
628 

444 
542 
637 

6454 
6551 
6646 

6464 
6561 
6656 

474 
57i 
665 

6484 
6580 
6675 

6493 
6590 

6684 

6503 

6599 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

1       3 
i       3 
1       3 

4  5  6 
4  5  6 
4  5  6 

7  8  9 
7  8  9 
7  7  8 

721 
812 
902 

730 
821 
911 

6739 
6830 
6920 

6749 
6839 
6928 

758 
848 
937 

6767 
6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 

6893 
6981 

1       3 

i       3 
1        3 

455 

6  7  8 

445 

6  7  8 

50 

990 

998 

7007 

7016 

024 

7033 

7042 

050 

7059 

7067 

*        3 

345 

6  7  8 

51 

52 
53 

076 
1  60 
243 

084 
168 
251 

7093 
7177 
7259 

7101 

7185 
7267 

no 
193 

275 

7118 
7202 
7284 

7126 
7210 
7292 

135 
218 
300 

7M3 
7226 
7308 

7152 
7235 
7316 

i        3 

I          2 
I          2 

345 
345 
345 

678 
6  7  7 
667 

54 

324 

332 

7340 

348 

356 

7364 

7372 

380 

7388 

7396 

122 

3  4  5 

6  6  7 

LOGARITHMS. 


57 


55 

0 

74CM 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

7412 

7419 

7427 

7435 

7443 

745i 

7459 

7466 

7474 

I  2  2 

345 

5  6  7 

66 
57 
58 

74«2 
7559 
7634 
7709 
7782 
7853 

7490 
7566 
7642 

7497 
7574 
7649 

7505 
7582 
7657 

7513 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

753t> 
7612 
7686 

7543 
7619 
7694 

755' 
7627 
7701 

I  2  2 

5  6  7 
5  6  7 

I  I  2 

344 

59 
60 
61 

7716 
7789 
7860 

7723 
7796 

7868 

773i 
7803 

7875 

7738, 
7810 

7882 

7745 
7818 
7889 

7752 
7825 
7896 

7760 
7832 
7903 

7767 

7839 
7910 

7774 
7846 

79T7 

112 
I  I  2 
112 

344 
344 
344 

5  6  7 
5  6  ( 
5  6  6 

62 
63 
64 

7924 

7993 
8062 

7931 
Sooo 
8069 

7938 
8007 
8075 

7945 
8014 
8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
8055 
812? 

I  1  2 
112 

334 
334 

5  6  6 
5  5  6 
5  5  * 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

112 

334 

5  S  C 

66 
67 
68 

8195 
826) 
8325 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

112 
(  I  2 
I  I  2 

334 
3  3  4 
334 

5  5  6 
5  5  6 
4  5  6 

69 
70 
71 

8388 
8451 
S$I3 

8395 
8457 
8519 

8401 
8463 

8525 

8407 
8470 
853i 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 

8555 

8439 
8500 
8561 

8445 
8506 

8567 

I  I  2 

2  3 

456 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

I  I  2 
112 
112 

2  3 

2  3 

2  3 

435 
455 
455 

75 
"76 
77 
78 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

I  I  2 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

885-9 
8915 
897) 

112 
112 

233 
233 

455 

445 

79 
80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 

9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 
9180 
9232 
9284 

9025 
9079 
9133 

I  I  2 
112 
I  I  2 

233 
233 

445 
445 

82 
83 
84 

9138 
9191 
9243 

9143 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9*59 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 

9279 

9186 
9238 
9289 

112 
I  I  2 
112 

233 
233 
233 

445 

445 
445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

I  1  2 

233 

445 

86 
87 
88 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

936o 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

1  I  2 

0  1  I 
Oil 

233 

2  3 

2  3 

4  4  5 
344 
344 

89 
90 
91 

9494 
9542 
959° 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

95i8 
9566 
9614 

9523 
957" 
9619 

9528 
9576 
9624 

9533 
958i 
9628 

953& 
9586 

9633 

Oil 
Oil 
Oil 

2  3 

2  3 

2  '3 

3  4  4 
344 
3  4  4 

92 
93 
94 

9638 
9685 
9731 

9643 
9689 
9736 

9647 
9694 
974i 

9652 
9699 
9745 

9657 
9703 
9750 

9661 
9708 
9754 

9666 
97*3 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

Oil 
Oil 
Oil 

2  3 
2  3 

2  -  3 

344 
344 
3  4  4 

95 

9777 

9782 

9786 

9791 
9836 
9881 
9926 

9795 

9800 

9805 

9809 

9814 

9818 

0  1.  I 

2  3 

3  4  4 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 
9921 

9841 
9886 
9930 

9845 
9890 
9934 

9850 
9894 
9939 

9854 
9899 

9943 

9859 
9903 
9948 

9863 
9908 
9952 

0  I  I 
Oil 
Oil 

2  3 
2  3 
2  3 

344 

344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

999  1 

9996 

0  I  1 

223 

334 

58 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

ooooo 

043 

087 

130 

173 

217 

260 

303 

346 

389 

101 
102 
103 

432 
860 
oi  284 

475 
903 
326 

5i8 
945 
368 

56i 
ciSS 
410 

604 
030 
452 

647- 
072 

494 

689 
"5 
536 

732 
157 
578 

775 
199 
620 

817 
242 
662 

104 
105 
106 

703 

02  119 

531 

745 
1  60 
572 

787 
202 
612 

828 
-43 
653 

870 
284 
694 

912 
325 

735 

953 
366 

77* 

995 
407 
816 

036 

449 
857 

078 
490 
898 

107 
108 
109 

938 
03342 

743 

979 

383 
782 

019 

423 
822 

060 
463 

862 

100 

503 
902 

.41 

543 
94i 

181 

583 
981 

222 
623 
021 

262 
663 

060 

302 
703 

100 

To  find  the  logarithm  of  a  number:  First,  locate  in  the 
table  the  mantissa  which  lies  in  line  with  the  first  two  figures  of  the 
number  and  underneath  the  third  figure,  then  increase  this  mantissa 
by  an  amount  depending  upon  the  fourth  figure  of  the  number  and 
found  by  means  of  the  interpolation  columns  at  the  right;  secondly, 
determine  the  characteristic,  or  the  exponent  of  that  integer  power 
of  10  which  lies  next  in  value  below  the  number;  for  example, 
log  600-  0.7782  t  2. ;  -log  73.46=  0.8661  +  1. ; 

log  .006=  0.7782-3. ;  log  .7346=  0.8661  - 1. ; 

log  6.003=  0.7784  +  0. ;  log  7349=  0.8662  +  3. 

The  logarithm  of  a  product  of  two  or  more  numbers  is  the  sum  of 
the  logarithms  of  its  factors;  for  example, 

log.  (.0821  X  463.2)  =  (0.9143  -  2.)  +  (0.6658  +  2.)  =  0.5801  +  1. 
The  logarithm  of  a  quotient  is  the  difference  between  the  logar- 
ithms of  the  dividend  and  divisor;  for  example, 

log.  (.5321-*- 916)=  (0.7260-1.)  -  (0.9619 +  2.)  =  0.7641 -4. 
The  logarithm  of  a  power  or  root  of  a  number  is  the  exponent 
times  the  logarithm  of  the  number;  for  example, 

log  V/:863)T=3/2X(0.9360-1.)  =  0.9040-1. 

To  find  the  number  from  its  logarithm:  Locate  in  the  table 
the  mantissa  next  less  than  the  given  mantissa,  then  join  the  figure 
standing  above  it  at  the  top  of  the  table  to  the  two  figures  at  the 
extreme  left  on  the  same  line  as  the  mantissa,  and  finally  to  these 
three  join  the  figure  at  the  top  of  the  interpolation  column  which 
contains  the  difference  between  the  two  mantissae.  In  the  four- 
figure  number  thus  found,  so  place  the  decimal  point  that  the 
number  shall  be  the  product  of  some  number,  that  lies  between 
1  and  10,  by  a  power  of  10  whose  exponent  is  the  characteristic 
of  the  logarithm.  For  example, 

antilog  (0.6440 +  3)  =  4405; 
antilog  (0.3069  -  2)  =  .02027. 

Caution.  In  adding  and  subtracting  logarithms  it  is  well  to 
remember  that  the  mantissa  is  always  essentially  positive  and  may 
or  may  not  therefore,  have  the  same  sign  as  its  characteristic. 


INDEX 


Ampere,  defined        -         -  24 

Ammeter,  tested       -          -  27 

Bifilar  suspension     -  17 

Bridge,  Wheatstone,  used  29 
Calibration,     relative     of 

Galvanometer       -         -  37 
Capacity,    specific    induc- 
tive, determination  of  -  47 
Capacities,  comparison  of  47 
Cell,  standard,  use  of        -  36 
Coefficient    of    resistance, 
temperature,       deter- 
mined                               -  31 

Commutator,  use  of          -  25 

Conductivity,  specific       -  42 

Conductivity,  molecular  -  43 

Connections,  electrical      -  23 

Current,  unit,   defined     -  23 
Demagnetization,  method 

of  -  20 
Earth  inductor  -  40 
Electrolyte,  resistance  of  -  42 
Electromotive  forces,  com- 
parison of  -  -  34 
End-on  position  -  -  15 
Experiments,  list  of  5 

Field,  electric,  plotting    -  10 

Magnetic,  location  of    -  7 

Plotting         -                   -  •  9 
Galvanometer,    ballistic, 

calibration  of                  -  19 

Tangent,  constants  of  -  27 

"    reduction  factor  of  26 

theory  of                  -  23 

used  as  voltmeter           -  45 

"H"  defined  7 
relative     determination 

of                    -         f       -  14 
absolute    determination 

of                                       -  15 

Heating  devices,  electrical  47 

Hot-plate,  efficiency  of      -  47 

Induction,  magnetic           -  19 
Inclination,  or  dip,  angle 

of                                       -  41 

Law,  tangent,  test  of         -  24 


Lines    of    force,    direction 

of  magnetic  7 

intensity  of  magnetic    -  7 

tracing  of  electric           -  10 

tracing  of  magnetic       -  9 

Magnetic  force,   direction 

of,  defined    -                   -  7 

intensity  of,  denned      -  7 

laws  of                              -  13 

Magnetometer,     mirror, 

used     -                             -  17 

Magnets,  care  of  9 

Method,  Carey  Foster  30 

of  oscillations        -         -  14 

Kohlrausch's                    -  43 

Poggendorff's        -         -  34 

Moment  of  magnet            -  9 

Permeability,  defined        -  8 

Poles,  of  magnet,  defined  8,  23 

Polarization,  in  Leclanche 

cell        -                             -  45 
Postoffice  bridge,  used      -  34 
Reduction  factor,  of  gal- 
vanometer, defined       -  24 
Reference  books  6 

Resistance,  absolute  deter- 
mination of  -  32 
boxes,  care  of  -  23 
comparison  of  -  -  30 
defined  -  32 
specific,  determined  -  31 

Robison  magnet        -          -  15 

Solenoid,     field     intensity 

for        -                             -  19 

Suspension,  bifilar,  used  -  17 

Tables,  of  logarithms        -  56 

physical         -                    -  49 

of  sines  and  tangents    -  52 

Thermo     electromotive 

force.     -          -         -         -  37 

Tangent    galvanometer, 

constants  of                     -  27 

reduction  factor  of         -  26 

Voltameter,  copper,  used  -  26 

Water,  equivalent  of  heat- 
ing coil          -                   -  33 
Wheatstone  bridge,  used  -  29 


PHYSICAL   MEASUREMENTS 


MINOR 


PART  IV.    SOUND  AND  LIGHT 
1917 


PHYSICAL   MEASUREMENTS 

A  LABORATORY  MANUAL  IN  GENERAL  PHYSICS 
FOR  COLLEGES 

In  Four  Parts 

BY 
RALPH  S.  MINOR,  Ph.  D. 

Associate  Professor  of  Physics,  University  of  California 


PART  IV 
SOUND  AND  LIGHT 


In  collaboration  with 
Raymond  B.  Abbott,  M.  S. 

Instructor  in  Physics,   University  of  California 


BERKELEY,  CALIFORNIA 
1917 


Copyrighted  in  the  year  1917  by 
Ralph  S.  Minor 


Wetzel  Bros.  Printing  Co. 

21  10  Addiwn  Street 

Berkeley.  Calif. 


LIST  OF  EXPERIMENTS 

Sound  Page 

81.  Velocity  of  Sound  in  Solids  5 

82.  The  Natural  Scale.     Harmony  -       8 

83.  Longitudinal  Vibration  of  Wires  -       9 

Light 

The   Convention   of   Signs   for   Mirrors,  Single 

Refracting  Surfaces  and  Lenses  11 

Path  of  Any  Oblique  Ray  12 

Spherical  Mirrors      -  12 

The  Spherometer  13 

91.  Radius  of  Curvature.     By  Reflection  -     15 

92.  Refraction  at  a  Single  Plane  Surface  -     17 

93.  The  Refraction  Trough 

Refraction  at  a  Single  Spherical  Surface  20 

A  Study  of  the  Model  Eye  21 

94.  The  Spectrometer.  Measuring,  the  Angles  of  a 

Prism  24 

95.  The  Spectrometer.      Index  of   Refraction  of   a 

Prism  28 

96.  Refractive  Index  by  Total  Reflection 

Pulfrich's  Refractometer  30 
Thin  Lenses       -  -     33 
Direct  Measurement  of  Focal  Length  of  Con- 
vex Lenses  -     33 

97.  Refraction  Through  a  Single  Thin  Lens  33 

98.  Thick  Lenses  36 

99.  Astronomical  Telescope     -  -     40 

100.  The  Spectroscope  44 

101.  Spark  Spectra  of  Metals   -  47 

102.  Wave-Length  of  Sodium  Light 

Newton's  Rings  -     48 

103.  The  Diffraction  Grating 

The  Grating  Constant  50 

104.  Measurement  of  Wave  Length 

With  Spectrometer  and  Grating  -     52 

105.  Reflection  of  Light  on.  Transparent  Substances  -     53 


REFERENCE  BOOKS 

Clay :  Treatise  on  Practical  Light. 

Duff:  Text  Book  of  Physics.     (Fourth  Edition). 

Kdser:  Light  for  Students. 

Ganot:  Text  Book  of  Physics.     (18th  Edition.) 

Kimball:  College  Physics. 

Kaye  and  Laby:    Physical  and  Chemical  Constants. 
Landolt  and  Bernstein:  Physical  and  Chemical  Tables. 
Smithsonian  Institute:  Physical  Tables. 


SOUND 


81.     VELOCITY  OF  SOUNDS  IN  SOLIDS. 
Kundt's  Method. 

Kundt's  method  of  determining  the  velocity  of  sound  in 
solids  and  gases  consists  in  setting  up  stationary  waves 
in  a  horizontal  glass  tube  by  means  of  the  longitudinal 
vibrations  of  a  rod,  the  form  of  the  stationary  wave  being 
made  evident  by  the  recurring  patterns  into  which  lyco- 
podium  powder,  or  other  dust,  is  thrown  when  the 
resonance  is  sufficiently  vigorous.  A  great  variety  of 
forms  will  be  observed  in  these  dust  figures,  but  the 
exact  conditions  which  result  in  any  pattern  are  difficult 
to  define  as  much  depnends  upo  the  manner  of  rubbing 
the  vibrating  rod,  the  size  of  the  dust  particles,  the  dry- 
ness  of  the  walls  of  the  tube,  the  amount  of  dust  present, 
the  length  of  the  glass  tube,  the  length  of  the  vibrating 
column  of  air,  and  the  space  between  the  disk  on  the 
end  of  the  rod  and  the  glass  tube. 

In  every  pattern,  however,  the  distance  between  corres- 
ponding points  of  adjacent  waves  will  be  exactly  one-half  a 
wave  length  of  the  sound  in  air. 

It  should  be  particularly  noted  that  irregularities  may 
occur,  in  some  patterns,  both  near  the  closed  end  of  the 
tube  and  near  the  end  of  the  vibrating  rod,  so  that  in 
general,  measurements  should  not  include  the  first  half 
wave  length  at  either  end. 

Figure  t  is  the  pattern  usually  obtained.  Figure  2  is 
often  obtained  when  the  length  of  the  vibrating  column  is 


8  THE  NATURAL  SCALE.     HARMONY  [82 

Additional  Exercise. 

Determine  the  velocity  of  sound  in  tubes  of  various 
diameters  presenting  your  results  in  the  form  of  a  plot 
showing  the  relation  between  the  velocity  of  sound  in 
air  and  the  diameter  of  the  tube. 

82.     THE  NATURAL  SCALE.     HARMONY. 

The  sonometer  should  be  provided  with  two  strings  of 
the  same  material  and  diameter.  The  strings  should  be 
tuned  in  unison. 

(a)  Place  a  bridge  under  one  of  the  strings  and  grad- 
ually change  the  length  of  the  vibrating  portion  until 
you  have  secured  a  tone  which  in  your  judgment  sounds 
well  when  heard  with  the  string  whose  length  has  not 
been  changed. 

Shorten  the  vibrating  part  still  further  until  you  have 
a  second  tone  which  goes  well  with  the  fundamental 
tone.  Strike  these  three  notes  in  rapid  succession.  They 
form  a  major  triad.  Together  with  the  octave  of  the  first 
note  they  form  a  major  chord. 

From  your  measurements  determine  the  ratio  of  the 
vibration  rates  for  these  four  notes. 

Place  a  bridge  under  the  first  string  so  that  it  gives 
the  second  of  the  new  tortes  found  and  see  if  with  this 
tone  as  your  starting  point  you  can  find  two  other  notes 
which  sound  well  with  it.  That  is,  find  another  major 
triad.  Find  the  ratio  of  their  vibration  rates. 

Starting  with  the  octave  of  your  first  fundamental 
tone  find  another  triad  by  increasing  the  length  of  the 
string  until  a  pleasing  combination  similar  to  the  two  first 
found  is  the  result. 

Find  the  ratios  of  the  vibration  rates  of  the  eight  tones 
in  terms  of  the  fundamental,  i.  e.  the  tone  given  out  by 
the  entire  string. 


82-83]  THE  NATURAL  SCALE  9 

Strike  these  tones  in  succession,  in  order  of  increasing 
vibration  rate  and  you  have  the  natural  or  diatonic  scale. 

Why  is  this  scale  not  used  on  the  piano? 

(b)  In  order  to  determine  the  physical  basis  for  har- 
mony, test  the  triads  separately  for  sympathetic  vibra- 
tions of  the  second  string  for  any  of  the  positions  found 
when  the  two  sound  well  together.  In  what  way  must 
the  first  string  be  vibrating  in  order  that  such  sympa- 
thetic vibrations  may  be  set  up?  Is  the  intensity  of  the 
sympathetic  vibration  greater  in  the  case  of  two  notes 
of  a  triad  than  for  two  notes  not  so  related.  If  possible 
repeat  this  test  on  a  piano. 

After  considering  the  overtones  possible  on  a  string  in 
transverse  vibration  together  with  your  experimental 
work  make  a  statement  as  to  the  secret  of  harmony. 

83.     LONGITUDINAL  VIBRATION  OF  WIRES. 

The  object  of  this  experiment  is  to  determine  Young's 
modulus  for  the  material  of  a  wire  from  measurements  of 
the  wave-length  of  its  longitudinal  vibrations  and  its  density. 

A  wire  about  ten  meters  long  is  clamped  at  each  end 
by  means  of  two  vises.  The  attachment  at  one  end  being 
to  the  movable  jaw  of  the  vise  so  that  the  tension  may 
be  changed.  The  length  of  the  vibrating  part  is  varied 
by  means  of  a  movable  bridge. 

The  points  at  which  the  wire  meets  the  vise  and  the 
bridge  are  nodes,  and  the  distance  /  between  two  consecu- 
tive nodes  is  one  half  of  a  wave-length  (for  the  funda- 
mental). Therefore  the  velocity  v  =  2ln  =  V  E  /  d,  from 
which  Young's  modulus  E  may  be  found. 

(a)  With  the  wire  under  slight  tension  set  up  longi- 
tudinal vibrations  in  it  by  rubbing  it  with  the  fingers, 


10  LONGITUDINAL  VIBRATION  OF  WIRES  [83 

which  have  been  touched  with  powdered  rosin.  At  the 
same  time  sound  a  tuning  fork  of  known  frequency,  and 
adjust  the  vibrating  length  of  the  wire  until  it  gives  the 
same  note.  Approach  the  position  of  unison  from  the 
side  of  too  great  a  length  and  again  from  the  side  of  too 
small  a  length  and  take  the  mean  as  the  value  of  /. 

The  formula  given  above  states  that  the  velocity  of  a 
longitudinal  vibration  is  independent  of  the  tension  of 
the  wire.  If  this  is  the  case  the  pitch  of  the  note  should 
not  be  altered  when  the  tension  of  the  wire  is  varied  with- 
in wide  limits  (while  the  length  is  kept  constant).  Try 
this  experiment,  taking  care  not  to  stretch  the  wire  beyond 
the  elastic  limit  of  its  material. 

(6)  Determine  the  density  of  the  wire  from  the  sample 
furnished  by  any  method  you  select. 


(c)     Calculate   Young's   modulus  for  the    material   of 
the  wire. 


LIGHT 


The  Convention  of  Signs  for  Mirrors,  Single  Refracting 
Surfaces  and  Lenses. 

In  writing  the  formulae  which  show  the  relation  be- 
tween the  object  distance  u,  the  image  distance  v,  and 
the  constants  of  mirrors,  single  refracting  surfaces  and 
lenses  the  following  convention  of  signs  will  be  used  in 
the  manual  and  also  in  the  lecture  course  given  in  con- 
nection with  the  laboratory  work: 

1.  Distances  measured  over  a  path  actually  traveled  by 
the  light  are  called  positive,  +;    distances  measured  over  a 
projected  path  of  the  ray  are  called  negative,  — .     The  ray 
being  the  normal  to  the  wave-front. 

This  applies  to  the  object  distance  u,  the  image  dis- 
tance v,  and  the  focal  length  /.  In  accordance  with  this 
convention  the  distance  from  the  mirror  or  lens  to  a  real 
object,  a  real  image,  or  a  real  focus,  is  always  positive; 
negative  values  always  mean  that  the  object,  image  or 
focus,  is  virtual. 

2.  The  radius  of  curvature  of  a  surface  which  tends  to 
produce  a  real  focus  is  called  positive;  one  which  tends  to 
produce    a   virtual   focus   is    negative.      Accordingly   the 
radius  of  a  concave  mirror  is  positive,  since  it  brings  light 
to  a  real  focus,  while  the  radius  of  curvature  of  a  concave 
refracting   surface   is   negative,    as   it   renders    a   parallel 
beam   divergent.     Both   radii  of  curvature  of  a  double 
convex  lens  are  positive;  both  radii  of  curvature  of  a 
double  concave  lens   are  negative. 

This  convention  of  signs  places  the  real  physical  quantity  and 
the  positive  sign  together.  It  has  also  the  practical  advantage  of 
being  the  one  uniformly  used  by  opticians,  manufacturers  and 
dealers.  Any  convergent  system  has  a  positive  focal  length.  A 
divergent  system  has  a  negative  focal  length. 


12 


THE  CONVENTION  OF  SIGNS 


1  1 

For  a  mirror      ~^~  ~    ~ 


(i) 


For  a  single  refracting  surface  —  +  —  =  — - — 


(2) 
(3) 


For  a  lens   4~  +  ~T  =  ^n~1 )  ^~7T  +  ^  =  ~J~ 

For  a  combination  of  two  thin  lenses  in  contact,  whose 
focal  lengths  are  /j,  and  f,,  respectively,  the  focal  length 
of  the  combination  F,  is  given  by  the  formula 

111  ,  . 

Tr+TT^-Tr  (4) 


Path   of   Any    Oblique   Ray   Through   a    Single   Thin   Lens. 

Given  the  focal  points  FI  and  F2  and  the  optical  center 
0.  For  any  oblique  ray  AB  draw  a  parallel  to  AB  from 
the  optical  center  to  its  intersection  with  the  second 


0 


Fig.    5. 

Path  of  any  oblique  ray. 


focal   plane   at  C,   then   the   refracted   ray   is  BC,   since 
parallel  rays  intersect  in  the  focal  plane. 


Spherical  Mirrors. 

The  following  simple  optical  methods  are  useful  in 
determining  the  radius  of  curvature  of  spherical  mirrors 
or  lens  surfaces: 


THE  SPHEROMETER  13 

Concave  Mirrors. 

1.  Place  the  concave  surface  in  the  path  of  a  beam  of  parallel 
light — sunlight,  light  rendered  parallel  by  means  of  a  lens,  or  light 
from  a  distant  object — tilt  it  slightly  and  find  the  reflected  image 
with  a  card.     When  sharply  in  focus  the  distance  from  the  reflect- 
ing surface  to  the  image  will  be  the  focal  length,  or  one-half  the 
radius  of  curvature. 

For  an  object  not  very  distant,  the  radius  of  curvature  could 
similarly  be  determined  by  measuring  both  object  and  image 
distances. 

2.  Turn  the  concave  surface  to  be  tested  toward  an  illuminated 
aperture  in  a  white  screen.     A  slit  with  a  single  cross-hair  will  be 
found  convenient.     Adjust  until  the  image  is  seen  on  the  screen 
beside  the  slit.     The  distance  from  the  reflecting  surface  to  the 
screen  will  then  be  equal  to  the  radius  of  curvature. 

Convex  Mirrors. 

1.  Use  a  convergent  lens  to  form  a  real  image  of  a  narrow  slit 
in  a  white  screen.  Adjust  a  needle,  using  the  parallax  test,  so  that 
it  coincides  with  the  image.  Now  place  the  convex  surface  to  be 
tested  between  the  image  and  the  lens  (convex  side  toward  the 
lens)  and  adjust  until  an  image  of  the  slit  is  formed  on  the  white 
screen,  beside  the  slit.  In  this  case  the  distance  between  the  needle 
and  the  convex  surface  is  equal  to  its  radius  of  curvature. 

The  Spherometer. 

The  spherometer  furnishes  an  easy  mechanical  method 
for  determining  the  radius  of  curvature  of  spherical 
surfaces.  It  is  used  to  best  advantage  with  surfaces  of 
medium  curvature. 

A.     The  Ring  Spherometer. 

A  rotating  arm  is  provided  which  slips  over  the  point 
of  the  spherometer  screw.  By  means  of  this  the  upper 
brass  ring  should  be  adjusted  so  that  its  axis  coincides 
with  the  axis  of  rotation  of  the  screw  and  then  clamped  in 
position  by  means  of  the  small  thumb-screws  underneath. 
The  internal  radius  of  this  ring,  which  is  the  perpendi- 
cular distance  from  the  axis  of  the  screw  to  the  inner  edge 
of  the  ring,  is  called  the  "Span,"  5,  of  the  spherometer. 


14 


THE  SPHEROMETER 


Fig.  6.     A  section  through  the  lens 
and  spherometer  ring. 


If  the  difference  in  the 
reading  of  the  spherometer 
screw  when  the  ring  is 
covered  with  a  piece  of 
plate  glass  and  when  a 
lens  of  radius  of  curvature 
R  is  placed  upon  it  be  d, 
see  figure,  then  it  may  be 
shown  that  the  radius  of 
curvature  is  determined 
from  these  measurements 
by  means  of  the  formula 


d) 


2d 


If  the  pitch  of  the  spherometer  screw  is  not  standard  it  should 
be  determined  by  measuring  with  the  micrometer  calipers  the 
thickness  of  a  piece  of  glass,  whose  thickness  has  been  determined 
in  terms  of  the  pitch  of  the  spherometer  screw. 

B.     The  Peg-Spherometer. 

The  peg-spherometer  consists'  of  a  tripod,  the  legs  of 
which  form  an  aquilateral  triangle,  with  the  screw  at  its 
center.  If  /  represents  the  average  distance  between  the 
legs  of  the  tripod  and  d  the  mean  distance  from  the  point 
of  the  screw  when  resting  against  the  surface  of  a  lens, 
to  the  plane  through  the  ends  of  the  spherometer  legs,  it 
may  be  shown  that  the  radius  of  curvature,  R,  is  given 


by  the  expression 


(2) 


C.     The  Aldis  Peg- Spherometer. 

With  the  Aldis  peg-spherometer,  in  which  the  points 
on  the  ordinary  spherometer  are  replaced  by  small 
spheres,  of  average  radius  r,  the  relations  are  very  similar, 
R  +  r  taking  the  place  of  R  in  equation  (2). 


91]  RADIUS  OF  CURVATURE  15 

91.     RADIUS  OF  CURVATURE 
By  Reflection. 

The  following  method  of  determining  the  radius  of 
curvature  of  spherical  reflecting  surfaces  is  particularly 
useful  in  measuring  surfaces  of  small  radius,  as  well  as 
surfaces  inaccessible  to  measurement  by  other  methods. 

In  the  figure  DCE  is  a  section  of  the  reflecting  surface 
whose  center  is  0.  A  and  B  are  two  objects  and  A'  and 
B'  are  their  images. 


Fig.  7.     Illustrates  the  method  of  finding  the  radius  of  curvature  by  reflection. 
Let  P  C  =d,  P1C  =  -x,  O  C  =  -r,  A  B  =  L,  A1B>  =  /. 

112  r  d 

Then  -—  --  —  --      .-.    *=  -  -  —  n\ 

d          x  r  r  +  2d 

The  first  equation,  however  may  be  written 


—  r-  +  -  —  ---   From  which  follows 
d          r          x          r 


r  +  d 


From  the  figure        -=-  =<  — ,  .-.         -— -  =  JL 

L         r  +  d  L  d 


16  RADIUS  OF  CURVATURE  [91 

The  radius  of  curvature,  r,  of  a  mirror  (or  lens)  may 
thus  be  determined  from  measurements  of  the  distance 
apart,  L,  of  two  objects  and  the  distance,  /,  between  their 
images  formed  by  the  mirror  when  d  centimeters  away, 
To  measure  the  distance  /  between  the  images,  a  traveling 
telescope  should  be  placed  midway  between  the  objects 
A  and  B  with  its  axis  parallel  to  the  axis,  OP,  of  the 
mirror.  The  objects,  A  and  B,  as  well  as  the  telescope 
and  the  mirror  should  be  in  a  common  horizontal  plane. 

In  finding  the  radius  of  curvature  of  a  lens  surface  it 
must  be  remembered  that  each  surface  of  the  lens  will 
form  a  pair  of  images.  These  may  be  distinguished  by 
remembering  that  the  images  produced  by  a  convex  re- 
flecting surface  are  always  erect.  The  telescope,  however, 
inverts  them.  It  should  be  noted  that  formula  (2)  does 
not  hold  for  the  images  formed  by  the  second  surface  of  a 
lens,  as  the  light  which  strikes  it  has  suffered  refraction 
at  the  first  surface. 

To  find  the  radius  of  curvature  of  a  lens. 

(a)  Measure  with  a  meter  rod  the  distance,  d,  meas- 
uring from  the  line  joining  the  two  objects  to  the  reflect- 
ing surface;  and  the  distance,  L,  between  the  centers  of 
the  objects  A  and  B,  (slits  in  a  screen,  illuminated  in 
turn  by  a  frosted  bulb  incandescent  lamp). 

(6)  To  find  /  set  the  cross-hairs  of  the  eyepiece  on 
one  of  the  images  and  read  the  scale  and  head  of  the 
micrometer  screw.  Make  several  settings,  always  turn- 
ing the  screw  in  the  same  direction  in  coming  to  the 
final  position,  and  take  the  average.  Then  turn  the  screw 
until  the  cross-hairs  coincide  with  the  other  image,  take 
several  readings,  averaging  them  as  before,  being  careful 
in  making  any  setting  to  turn  the  screw  in  the  same 
direction  as  in  the  first  case  in  coming  to  the  position  of 
coincidence  between  image  and  cross-hairs. 


91-92]    REFRACTION  AT  A  SINGLE  PLANE  SURFACE     17 

(c)     Calculate  the  value  of  r  from  equation  (2). 
What  modification  in  equations  (1)  and  (2)  would  be 
introduced  if  the  reflecting  surface  were  concave? 

92.     REFRACTION  AT  A  SINGLE  PLANE  SURFACE. 

The  object  of  this  experiment  is  to  observe  the  visual  effects 
due  to  the  astigmatism  which  results  when  spherical  waves 
are  refracted  at  a  plane  surface,  and  to  plot  the  caustic  curve 
of  the  refracted  wave. 


AIR 


AIR 


WATER 


Fig.  8. 

In  the  figure  let  the  point  0  be  a  source  in  water  from 
which  spherical  waves  emanate  passing  into  air  across  the 
plane  surface  A  B.  The  dotted  line  aa  represents  the 
position  an  advancing  spherical  wave  would  have  reached 
at  a  given  instant  if  it  had  not  passed  out  of  the  water 
into  the  air,  and  the  line  bbb  represents  the  actual  wave  in 
air.  This  wave  bbb  is  not  spherical  as  we  see  by  compari- 
son with  the  arc  C  V;  that  is,  refraction  at  a  plane  surface 
is  subject  to  spherical  aberration.  Furthermore,  a  small 
portion  of  the  wave  at  dd  has  two  radii  of  curvature;  or 
in  other  words,  the  pencil  of  rays  near  dd  is  an  astigmatic 
pencil,  that  is,  oblique  refraction  at  a  plane  surface  is 
subject  to  astigmatism.  The  intersection  of  the  wave  bb 
with  the  plane  of  the  paper  forms  at  dd  the  arc  of  a  circle 


18       REFRACTION  AT  A  SINGLE  PLANE  SURFACE      [92 

whose  center  is  at  K.  Suppose  a  plane  to  be  passed 
through  the  line  L  K  d  perpendicular  to  the  plane  of  the 
paper.  The  intersection  of  the  wave  bb  with  this  plane 
at  dd  is  the  arc  of  a  circle  whose  center  is  at  L. 

Suppose  a  person  to  look  at  the  point  O,  allowing  the 
pencil  of  rays  dd  to  enter  the  two  eyes.  If  the  head  is  held 
erect  so  that  the  eyes  lie  in  a  horizontal  line  (perpendicular 
to  the  plane  of  the  paper)  then  the  curvature  of  the  por- 
tion dd  of  the  wave  front  about  the  vertical  axis  deter- 
mines the  location  of  0  as  perceived  by  the  two  eyes,  and 
the  point  0  appears  at  L.  If,  on  the  other  hand,  the  head 
is  held  on  one  side  so  that  the  two  eyes  lie  in  a  vertical 
line  (in  the  plane  of  the  paper)  then  the  curvature  of  the 
portion  dd  of  the  wave  front  about  a  horizontal  axis 
determines  the  location  of  0,  as  perceived  by  the  two 
eyes,  and  the  point  O  appears  at  K.  A  short  horizontal 
line  through  K  and  a  short  vertical  line  at  L  are  the  two 
focal  lines  of  the  astigmatic  pencil  dd. 

The  caustic  curve  0' K  is  the    locus    of    the    center 

of  curvature  of  the 
wave  front  bbb,  or  the 
caustic  curve  may  be 
defined  as  follows: 

Imagine  lines  to  be 
.  T  r    drawn  normal  to  the 

Fig.  9.  Showing  the  focal  lines    KK  andLL 
of  the  astigmatic  pencil  dddd.  Curve       bb       Sit       each 

point;  the  envelope  of  these  lines  will  be  the  caustic  curve. 

The  apparatus  consists  of  a  metal  strip  carrying  a  ver- 
tical scale  VH  and  a  horizontal  scale  A  B  in  a  jar  of 
water,  the  scale  A  B  lying  on  a  level  with  the  surface  of 
the  water.  The  lower  end  of  the  extension  H  0  is  turned 
upward  and  has  a  narrow  straight  edge  of  celluloid  in- 
serted in  the  end.  The  rest  of  this  extension  as  well  as 
the  inside  of  the  jar  is  painted  black  so  as  to  make  the 


92]      REFRACTION  AT  A  SINGLE  PLANE  SURFACE       19 

edge  0  clearly  visible  and  to  enable  one  to  see  clearly  the 
reflected  image  of  the  vertical  scale  in  the  water  surface. 
(a)     Note  that  the  polished  edge  O  appears  to  lie  against 
the  face  of  the  reflected  image  of  the  vertical  scale  when 
the  line  joining  the  observer's  eyes 
is  horizontal,  whereas  it  seems  very 
much   nearer  to  the  eyes  than   the 
reflected  image  of  the  vertical  scale 
when   the   line   joining   the    eyes   is 
vertical. 

To  determine  the  caustic  curve, 
look  at  the  polished  edge  O  with  one 
eye  placed  at  E,  in  the  plane  of  the 
scales  V  H  •-  AB,  and  record  the 
reading  on  the  reflected  image  of  the  vertical  scale  of 
the  point  L  where  the  polished  edge  appears  to  stand, 
and  record  the  corresponding  reading  of  the  point  K  where 
the  line  of  sight  crosses  the  horizontal  scale. 

Observe  the  readings  L  and  K  for  a  series  of  positions 
of  the  eye  E  covering  as  wide  a  range  as  possible. 

Measure  the  distance  A  0  from  the  top  of  the  scale 
A  B  to  the  edge  0. 


(b)  Make  a  plot  showing  the  scales  O  V  and  A  B  as 
axes,  lay  off  each  pair  of  observed  values  of  H  K  andH  L 
and  draw  each  straight  line  LK.  Add  to  the  plot  the 
point  O'  calculated  from  the  measured  distance  A  O  and 
the  refractive  index,  and  draw  the  totally  reflected  ray, 
then  draw  the  caustic  curve. 

State  Huygens'  principle.  Taking  the  refractive  index 
of  water  with  reference  to  air  as  4/3  find  by  graphical 
application  of  this  principle  a  wave  front  in  air  emanating 
from  a  point  source  below  a  plane  boundary  between  air 
and  water.  Show  by  comparison  that  this  wave  front  is 
not  spherical. 


20   REFRACTION  AT  A  SINGLE  SPHERICAL  SURFACE   [93 
93.     THE  REFRACTION  TROUGH. 

A.     Refraction  at  a  Single  Spherical  Surface. 

The  exercise  is  a  study  of  the  conjugate  relation  be- 
tween object  and  image  for  a  single  refracting  surface, 
together  with  a  determination  of  the  constants  (radius  of 
curvature,  refractive  index,  and  focal  length)  of  such  a 
surface.  The  apparatus  consists  of  a  long  trough  with  a 
thin  spherical  watch  glass  set  in  one  end,  so  that  the 
liquid  filling  the  trough  will  have  a  convex  surface  out- 
wards. The  refraction  of  the  watch  glass  is  to  be  neglected. 

(a)  Determine  the  radius  of  curvature  of  the  surface 
or  surfaces  to  be  used  by  any  of  the  methods,  described 
in  a  separate  exercise. 

(b)  Place  the  object  screen  on  the  air  side  of  the  sur- 
face  at   distances  from  the  surface   which  gives   a  real 
image  in  the  liquid.     Locate  this  real  image  by  means  of 
a  movable  screen  placed  in  the  liquid. 

Immerse  an  object  screen  in  the  liquid,  illuminating 
it,  if  necessary,  through  the  plate  glass  end  of  the  trough, 
place  the  screen  at  such  a  distance  from  the  surface  that 
a  real  image  is  formed  in  air.  Locate  the  image  with  a 
screen  and  eyepiece.  See  if  image  and  object  distances 
are  reversible  by  taking  an  object  distance  equal  to  one 
previously  used  and  find  the  image. 


(c)  Find  the  refractive  index  of  the  liquid  from  each 
of  the  observed  values  of  u  and  v. 

(d)  Show  that  for  a  single  refracting  surface 

n       J_       n  -  l 

V  u  r 


93]  THE  MODEL  EYE  21 

where  n  is  the  refractive  index  of  the  medium  into  which 
the  light  passes  relative  to  the  medium  containing  the 
object.  The  radius  of  curvature  of  a  convex  refracting 
surface  is  considered  positive,  the  radius  of  curvature  of 
a  concave  surface  is  considered  negative. 

Define  conjugate  points. 

The  first  principal  focal  distance  is  defined  as  the  value 
of  u  when  l/v  =  0,  i.  e.t  when  the  light  is  parallel  after 
leaving  the  surface.  The  second  principal  focal  distance 
is  defined  as  the  value  of  v  for  1  /  u  =  0,  i.  e.,  parallel 
light  is  incident  on  the  surface.  Calculate  the  first  and 
second  principal  focal  distances  for  the  surfaces  or  the 
surface  used,  assuming  the  object  in  air. 

Additional  Exercise  with  the  Refraction  Trough. 

Having  determined  the  constants,  (radii  of  curvature 
and  index)  of  a  convex  lens,  place  it  in  the  refraction 
trough  in  place  of  the  watch  glass  and  determine  its  focal 
length  on  the  water  side  from  measurements  of  conjugate 
foci.  Compare  this  value  with  the  calculated  value. 

B.     Study  of  a  Model  Eye.     Myopia,  Hypermetropia,  Astig- 
matism. 

The  refraction  trough  with  the  meniscus  is  to  represent 
the  optical  system  of  the  eye.  The  white  screen  corres- 
ponding to  the  retina,  and  the  refraction  of  the  meniscus 
corresponding  to  the  total  refraction  of  the  normal  eye. 
Abnormal  vision  in  this  model  eye  will  be  produced  by 
placing  a  single  lens  close  to  the  meniscus,  while  the 
correcting  glasses  will  be  placed  in  the  two  groves 
slightly  in  advance  of  the  first  one,  corresponding  to  the 
distance  at  which  eye  glasses  are  ordinarily  worn. 


22  THE  MODEL  EYE  [93 

As  a  test  object  an  incandescent  lamp  with  a  clear 
globe  or  a  small  arc  light  is  to  be  used.  It  should  be 
placed  behind  a  small  circular  opening  at  a  distance  of 
about  two  feet,  with  space  on  the  table  to  move  the 
lamp  about  a  foot  further  away  from  the  trough. 

(a)  Focus  the  image  sharply  on  the  white  screen 
with  the  lamp  about  two  feet  away  and  consider  the 
position  of  the  white  screen  thus  found  as  the  proper 
position  of  the  retina  for  the  normal  eye. 

(6)  The  Myopic  Eye.  In  the  myopic  eye  the  image 
of  a  distant  object  seen  without  accommodation  falls  in 
front  of  the  retina.  Either  because  the  eyeball  is  too 
long,  the  usual  case,  or  because  the  lens  is  too  strong. 
Bringing  the  object  nearer  will  throw  the  image  further 
back  and  hence  improve  vision.  A  person  with  myopic 
eyes  is  called  "near-sighted." 

Represent  the  myopic  eye  by  placing  a  lens  marked 
M  just  in  front  of  the  meniscus.  Note  the  change  in  the 
appearance  of  the  image.  Test  for  improvement  of  the 
image  by  moving  the  object  nearer  to  the  "eye."  Place 
the  object  in  its  original  position  and  find  a  correcting 
lens  which  will  bring  the  image  to  its  former  sharpness. 
Note  the  power  and  form  of  this  lens. 

(c)  The  Hypermetropic  Eye.  In  the  hypermetropic 
eye  the  image  of  a  distant  object  seen  without  accommo- 
dation falls  behind  the  retina.  Hither  because  the 
eyeball  is  too  short  or  the  refracting  system  is  too  weak. 
Vision  will  be  best  for  objects  furthest  away,  hence  the 
term  "far-sightedness."  Represent  the  hypermetropic 
eye  by  placing  a  lens  marked  H  in  front  of  the  meniscus. 
Test  for  improvement  of  the  image  by  moving  the 
object  away  from  the  eye.  Place  the  object  in  the 


93]  THE  MODEL  EYE  23 

original  position  and  find  the  lens  which  brings  the 
image  to  its  proper  focus  on  the  "retina."  Note  the 
power  and  form  of  this  correcting  lens. 

(d)  The  Astigmatic  Eye.  In  the  astigmatic  eye 
the  images  of  points  are  not  points  but  lines.  The 
effect  of  the  refracting  system  is  that  of  a  combination 
of  a  spherical  lens  with  a  cylinder.  The  power  of  the 
eye  is  greater  in  one  plane  than  in  a  plane  at  right  angles 
to  it. 

Represent  the  astigmatic  eye  by  placing  the  lens 
marked  A  in  front  of  the  meniscus.  Try  to  secure  im- 
provement of  vision  by  changing  the  position  of  the 
object;  and,  with  the  object  in  its  initial  position,  by 
the  use  of  various  correcting  glasses. 

What  is  the  relation  between  the  axis  of  the  correcting 
lens  and  the  plane  of  the  image  seen  without  this  lens? 

With  a  +4  cylinder  in  front  of  the  eye  study  the 
form  of  the  cone  of  light  when  the  wave  surface  is 
astigmatic. 


(e)  Draw  diagrams  showing  the  initial  and  final 
positions  of  the  object,  eye,  correcting  lenses,  and  retina, 
in  the  cases  studied  above. 


24  THE  SPECTROMETER  [94 

94.     THE  SPECTROMETER. 

Measuring  the  Angles  of  a  Prism. 

The  spectrometer  consists  of  a  graduated  circle, 
generally  fixed  in  a  horizontal  position,  to  which  is  at- 
tached a  collimator  and  a  telescope.  The  collimator 
consists  of  a  tube  containing  an  achromatic  lens  at  one 
end  and  a  vertical  slit  at  the  principal  focus  of  the  lens. 
The  movable  arm  carries  an  astronomical  telescope  which 
is  always  directed  towards  the  center  of  the  graduated 
circle.  The  position  of  the  telescope  with  reference  to 
the  fixed  circle  may  be  read  by  means  of  a  vernier. 

Above  the  center  of  the  graduated  circle  is  a  horizontal 
table,  called  the  table  of  the  spectrometer,  which  is  capa- 
ble of  rotation  about  the  vertical  axis  of  the  circle.  The 
position  of  the  circle  table  can  be  determined  by  means 
of  the  vernier. 

Adjustment  of  the  Spectrometer,    Using  a  Gauss  Eyepiece. 

1.  The  Eyepiece.     The  eyepiece  should  be  moved  in  or  out 
until  the  cross-hairs  are  distinctly  seen. 

2.  The  Telescope.     As  the  adjustment  secured  by  focusing  the 
telescope  on  a  distant  object  is  only  approximate  the  following 
method  of  focusing  the  telescope  for  parallel  light  should  be  used. 
Cover  the  objective  of  the  telescope  with  a  plane  mirror.     Place  a 
light  at  right  angles  to  the  axis  of  the  telescope  opposite  the  open- 
ing in  the  Gauss  eyepiece  so  that  the  cross-hairs  will  be  illuminated 
and  the  glass  plate,  inclined  at  45°  to  the  axis  of  the  tube,  sends 
some  light  down  the  barrel  of  the  telescope.     Move  the  tube  carry- 
ing the  eye-piece  and  cross-hairs  until  there  is  no  parallax  between 
the  cross-hairs  and  their  image,  formed  by  the  light  reflected  from 
the  plane  mirror.     The  cross-hairs  are  then  in  the  focal  plane  of 
the  telescope. 

Set  the  cross-hairs  at  45°  with  the  vertical. 

3.  1*he   Collimator.     Illuminate  the  slit,   turn  the  adjusted 
telescope  into  line   with   the   collimator   and  then,    while   looking 
through  the  telescope,  move  the  slit  in  or  out  till  there  is  no  parallax 
between  its  image  and  the  cross-hairs.     The  slit  is  then  in  the  focal 
plane  of  the  collimator  lens. 


94]  ADJUSTMENT  OF  THE  SPECTROMETER  25 

This  is  an  important  adjustment  since  when  the  slit  is  in  focus 
the  light  coming  from  the  collimator  has  a  plane  wave  front,  and 
the  curvature  of  the  wave-front  will  not  be  altered  by  the  intro- 
duction of  the  prism  or  by  any  lateral  displacement  of  the  prism. 

After  the  above  adjustments  have  been  made,  if  there  is  any 
difficulty  in  seeing  the  cross-hairs,  the  eyepiece  may  be  moved, 
but  not  the  cross-hairs  themselves. 

For  work  of  extreme  accuracy  the  axes  of  the  telescope 
and  collimator  must  lie  in  one  plane,  and  always  be  per- 
pendicular to  the  axis  about  which  the  telescope  rotates, 
and  the  faces  of  a  prism  or  the  plane  of  a  grating  placed 
on  the  spectrometer  table  should  be  parallel  to  this  axis. 

These  adjustments  require  patient  and  careful  manipu- 
lation, and  in  the  following  work  with  the  spectrometer 
the  student  may  assume  that  they  have  been  made. 
The  methods  of  adjustment  which  have  been  used  are 
given  below. 

To  adjust  the  Telescope  perpendicular  to  the  Axis  of  the 
Spectrometer:  Use  a  Gauss  eyepiece,  illuminating  the  cross-hairs 
with  some  convenient  source  of  light.  Place  a  plane-parallel  plate 
of  glass  upon  the  table  of  the  spectrometer  with  one  edge  parallel  to 
any  two  leveling  screws  (G  F.  See  figure  illustrating  first  method). 
Turn  the  table  so  that  the  glass  reflects  light  back  down  the  tele- 
scope and  adjust  the  table  until  the  reflected  image  of  the  cross- 
hairs is  seen  in  the  field  of  view.  If  the  cross-hairs  do  not  coincide 
vertically  with  their  image  correct  half  the  distance  by  adjusting 
the  table,  by  means  of  leveling  screw  H,  the  other  half  by  moving 
the  telescope.  Rotate  the  table  180°  until  the  reflected  image  is 
again  in  the  field.  If  the  two  images  coincide  for  both  positions  of 
the  table,  the  telescope  is  perpendicular  to  the  axis  of  the  instru- 
ment and  the  plane  of  the  glass  plate  is  parallel  to  this  axis. 

To  adjust  the  Collimator  perpendicular  to  the  Axis  of  the 
Spectrometer:  Illuminate  the  slit  and  rotate  the  telescope  until 
the  image  of  the  slit  is  seen  in  the  telescope,  adjust  the  collimator 
until  the  image  of  the  cross-hairs  on  the  collimator  slit  coincide  with 
the  intersection  of  the  cross-hairs  in  the  field  of  the  telescope.  The 
collimator  is  then  perpendicular  to  the  axis  of  the  instrument. 


26 


MEASURING  THE  ANGLES  OF  A  PRISM 


[94 


To  adjust  the  Faces  of  the  Prism  parallel  to  the  Axis  of  the 
Spectrometer:  Set  the  prism  on  the  spectrometer  table  with  the 
face  AC  at  right  angles  to  a  line  through  F  G.  The  inclination  of 
A  C  may  be  changed  by  turning  either  F  or  G.  Using  the  Gauss  eye- 
piece as  directed  above  adjust  A  C  until  the  cross-hairs  coincide 
with  their  image  in  the  field  of  view. 

Raising  or  lowering  H  will  simply  move  the  face  A  C  in  its  own 
plane  so  that  if  F  and  G  are  left  adjusted  the  face  A  C  may  be 
adjusted  by  turning  H  until  the  reflected  image  from  this  face  also 
coincides  with  the  cross-hairs.  Adjust  the  face  A  B  in  this  manner. 
All  three  faces  of  the  prism  are  then  vertical,  that  is,  parallel  to  the 
axis  about  which  the  telescope  (or  the  spectrometer  table)  rotates. 

(a)  Adjust  the  eyepiece,  telescope  and  collimator  of 
the  spectrometer  using  the  methods  outlined  above. 

Measure  by  either  one  of  the  following  methods  the 
three  angles  of  the  prism,  recording  the  number  of  the 
prism  and  distinguishing  the  angles  by  their  numbers  or 
letters.  Make  several  settings  for  each  angle. 

First  Method.  Keeping  the  Prism  fixed.  The  prism 
is  placed  on  the  table  of  the  spectrometer,  (see  figure)  with 

its  faces  A  B,  A  C  vertical 
while  the  parallel  beam  from 
the  collimator  falls  partly  on 
the  face  A  C,  and  partly  on 
the  face  A  B.  From  each  of 
these  faces  a  parallel  beam  is 
reflected,  and  if  either  of 
these  beams  falls  on  the  ob- 
jective of  the  telescope,  it 
will  be  brought  to  a  focus  on 
the  cross-hairs  of  the  latter. 
Turn  the  telescope  so  that 
the  image  of  the  slit  reflected 
from  one  face  coincides  with 

{fa^    intersection    of    the    tWO 


Fig.  11.     Illustrates  the  adjustment 
of  the  prism  and  the  first  method  of 

measuring  the  angles  of  a  prism. 


.  .,      , 

cross-hairs  and  read  the  ver- 


94] 


MEASURING  THE  ANGLES  OF  A  PRISM 


27 


niers.  Move  the  telescope  until  coincidence  is  again  ob- 
tained between  the  cross-hairs  and  the  image  reflected 
from  the  other  face  and  again  read  the  verniers. 

A  quick  method  of  finding  the  reflected  image  of  the 
slit  is  to  locate  the  reflected  beam  with  the  eye  first, 
then  on  swinging  the  telescope  in  front  of  the  eye,  the  slit 
will  be  found  in  the  field  of  view  on  looking  through  the 
telescope.  The  arm  should  then  be  clamped  fast,  the 
final  adjustment  always  being  made  by  means  of  the 
slow  motion  screw. 


(6)  Show  with  the  aid  of  a  diagram  that  the  difference 
of  the  two  readings  (the  angle  swept  out  by  rotation  of 
the  telescope)  is  twice  the  angle  of  the  prism  formed 
by  the  intersection  of  the  two  faces,  provided  the  incident 
light  is  parallel.  Tabulate  the  values  of  the  angles  of 
the  prism  given  by  your  data. 

Second  Method.  Keeping  the  Telescope  fixed.  Clamp 
the  telescope  so  that  its  axis  makes  any  convenient  angle 
with  the  axis  of  the  collimator.  Turn  the  table  carrying 
the  prism  until  the  image  reflected  from  one  face  of  the 
prism  coincides  with  the  intersection  of  the  cross-hairs, 

and  read  the  vernier  at- 
tached to  the  spectrom- 
eter table.  Turn  the  table 
until  coincidence  between 
the  cross-hairs  and  the  im- 
age of  the  slit  as  reflected 
from  another  face  of  the 
prism  is  obtained  and 
again  read  the  vernier. 
With  this  method,  the 

Fig.  12.     Illustrates  the  second  method  error    due    tQ    non_parallel_ 
of  measuring  the  angles  of  a  prism. 

ism  of  the  light  is  avoided. 


28  INDEX  OF  REFRACTION  OF  PRISM  [94-95 

To  avoid  the  refracted  images  which  are  sometimes 
present  in  measuring  the  angles  of  a  90°  prism  it  may  be 
found  convenient  to  shut  off  all  light  except  that  striking 
the  reflecting  surface  by  means  of  a  small  block  of  wood, 
or  piece  of  cardboard. 


(b)  Show  that  the  difference  in  the  two  readings  sub- 
tracted from  180°  is  the  angle  between  the  two  reflecting 
surfaces  and  hence  the  angle  of  the  prism. 

Present  in  tabular  form  the  values  of  the  angles  of 
the  prism  calculated  in  this  way  from  your  data. 

95.     THE  SPECTROMETER. 
Index  of  Refraction  of  a  Prism.     Dispersion. 

Use  sodium  light  as  a  source  and  adjust  the  spectrom- 
eter as  in  the  preceding  experiment. 

(a)  Measure  one  angle  of  the  prism  using  the  second 
method  described  above. 

(6)  With  the  prism  removed,  turn  the  telescope  to 
view  the  light  coming  directly  from  the  collimator,  and 
adjust  until  the  intersection  of  the  cross-hairs  coincides 
with  the  image  of  the  center  of  the  slit.  Displace  the 
telescope  and  make  another  setting  and  read  the  vernier. 
Repeat  several  times  and  use  the  average  of  the  results. 

Replace  the  prism  on  the  spectrometer  table,  in  the  path 
of  the  light  from  the  collimator,  so  that  the  light  is  re- 
fracted through  that  angle  of  the  prism  which  has  just  been 
measured.  Turn  the  prism  so  that  the  refracted  image 
appears  to  move  towards  the  direction  of  the  incident  light, 
continuing  the  motion  until  the  image  appears  to  stop. 
Further  turning  in  the  same  direction  will  now  cause  the 
image  to  move  away  from  the  direction  of  the  incident 
light.  Having  thus  roughly  found  the  position  of  mini- 
mum deviation,  turn  the  telescope  to  view  the  image  of 


95]  DISPERSION  OF  GLASS  PRISM  29 

the  slit.  Again  turn  the  prism  slightly,  first  one  way  and 
then  the  other,  seeking  to  find  the  position  of  the  prism 
for  which  turning  in  either  direction  causes  the  image 
to  move  away  from  the  direction  of  the  incident  light. 
Having  apparently  found  this  position,  carefully  set  the 
intersection  of  the  cross-hairs  on  the  center  of  the  slit 
image,  and  again  slightly  turn  the  prism  first  one  way 
and  then  the  other.  Any  small  motion  of  the  image 
toward  less  deviation  can  now  readily  be  detected,  the 
cross-hairs  serving  as  reference.  Having  the  prism  finally 
set  at  the  position  of  minimum  deviation  and  the  inter- 
section of  the  cross-hairs  coincident  with  the  center  of 
the  image  of  the  slit,  read  the  telescope  vernier. 

Displace  the  telescope  and  prism,  set  them  again  and 
take  a  second  reading.  Repeat  several  times  and  take 
the  average.  The  difference  between  the  minimum  devi- 
ation reading  and  the  direct  reading  gives  the  minimum 
deviation  D. 

(c)  Using  a  Bunsen  burner  and  a  salt  of  Lithium, 
determine  the  position  of  minimum  deviation  for  another 
wave-length. 


(d)     At  the  position  of  minimum  deviation  of  a  prism 
the  angle  of  incidence  and  the  angle  of  emergence  are 
equal.      From   this   relation   and   Snell's   law  show   that 
_   sin%(D  +  A) 

sin  ^A  where  A  is  the  angle  of  the  prism. 

Insert  the  values  A  and  D  previously  found  and  cal- 
culate the  indices  of  refraction  of  the  prism. 

Put  your  results  in  tabular  form  giving  the  number 
of  the  prism,  the  angle  used,  and  for  each  wave-length 
the  angle  of  minimum  deviation  and  the  index. 

The  index  of  refraction  of  any  transparent  liquid  can 
similarly  be  found  by  using  a  hollow  glass  prism. 


30  PULFRICH'S  REFRACTOMETER  [96 

96.     REFRACTIVE  INDEX  BY  TOTAL  REFLECTION. 

Pulfrich's  Refractometer. 

In  Pulfrich's  refractometer  the  light  is  usually  received 
at  grazing  incidence  on  the  boundary  surface  examined*. 
A  lens  throws  a  beam  of  light  on  the  lower  edge  of  the 
glass  cylinder  containing  the  liquid,  whose  refractive  index 
is  to  be  determined.  The  greater  the  angle  of  incidence 
a  ray  makes  with  the  boundary  line  between  the  liquid 


Fig.  13.     Pulfrich's 
Refractometer. 


and  the  glass  the  greater  will  be  the  angle  of  refraction. 
For  a  ray  just  parallel  to  the  boundary  (grazing  incidence) 
the  angle  of  refraction  reaches  a  maximum  value  called 
the  "critical  angle." 

If  now  the  angle  i  (see  figure)  be  the  angle  which  the 
ray  refracted  at  the  critical  angle  makes  with  the  normal 
to  the  vertical  face  of  the  prism  on  emergence,  then,  since 


*The  more  elaborate  forms  of  Pulfrich's  refractometer,  which 
are  fitted  with  a  cube  or  cylinder  of  glass  in  place  of  the  90  degree 
prism  here  shown,  also  permit  of  observations  by  the  method  of 
total  internal  reflection. 


96]  PULFRICH'S  REFRACTOMETER  31 

none  of  the  emerging  rays  can  make  a  smaller  angle,  the 
field  of  the  observing  telescope  must  be  dark  when  its 
axis  makes  a  smaller  angle  than  i.  The  angle  i  thus  deter- 
mined by  the  boundary  between  the  bright  and  dark 
regions  evidently  depends  upon  the  angle,  A,  and  refrac- 
tive index  of  the  glass  as  well  as  upon  the  refractive  index  of 
the  liquid.  If  the  first  two  are  known  the  refractive  index 
of  any  liquid  (provided  it  is  less  than  that  of  the  glass 
prism)  may  be  found  from  a  measurement  of  the  angle  i. 

(a)  Place  a  sodium  flame  about  10-15  centimeters 
from  the  lens  of  the  refractometer  so  that  it  is  seen  in  line 
with  the  cylinder  on  looking  through  the  lens.  Fill  the 
cylinder  with  water.  With  the  slow-motion  screw  released 
and  starting  with  the  axis  of  the  telescope  as  nearly 
vertical  as  possible  raise  it  slowly  while  looking  in  the 
field  for  a  yellow  band  of  light.  When  this  has  been 
found  tighten  the  slow  motion  screw  and  set  the  inter- 
section of  the  cross-hairs  on  the  boundary  line  between 
the  yellow  and  the  black,  remembering  that  it  is  the 
upper  edge  of  the  boundary  that  is  determined  by  the 
critical  angle  of  the  liquid  used. 

Take  several  readings,  moving  the  telescope  so  as  to 
approach  the  boundary  line  first  from  one  side  and  then 
from  the  other. 

Note  the  temperature  of  the  water  in  the  cylinder. 

To  find  the  angle  i  which  is  measured  from  the  normal 
to  the  prism  face  a  second  reading  should  be  taken  before 
the  prism  has  been  disturbed  in  any  way.  Set  the  tele- 
scope near  the  zero  of  the  scale,  illuminate  the  Gauss  eye- 
piece using  an  incandescent  lamp  set  at  right  angles  to 
the  axis  of  the  eyepiece  and  by  means  of  the  slow-motion 
screw  set  the  image  of  the  cross-hairs  (formed  by  light 
reflected  on  the  vertical  face  of  the  prism)  to  coincide 
with  the  cross-hairs  themselves,  or,  if  this  is  not  possible, 


32  PULFRICH'S  REFRACTOMETER  [96 

place  the  centers  of  the  two  crosses  in  the  same  horizon- 
tal line.  The  scale  reading  in  this  position  subtracted 
from  the  reading  first  taken  gives  the  true  value  of  the 
angle  i,  since  in  the  second  case  the  incident  and  reflected 
beams  coincide  and  the  axis  of  the  telescope  is  thus 
normal  to  the  vertical  face  of  the  prism. 

(6)  Determine  by  the  method  outlined  above  the 
angle  i  for  the  other  liquids  furnished,  noting  the  tem- 
perature in  each  case. 


(c)  If  A  be  the  angle  of  the  prism,  n  the  index  of 
refraction  of  the  glass,  and  N  the  index  of  refraction 
of  the  liquid  above  the  glass  and  the  angles  i,  r2,  n,  be, 
as  indicated  in  the  figure. 

From  the  geometry  of  the  figure  A  =  TI  +  r2.  (1) 

For  refraction  at  the  critical  angle  on  the  boundary 

liquid-glass  we  have        ~in  n    —  ~jj~      (2)     And  from  the 

refraction  on  emergence  a  third  relation      sin  rj    =n    (3) 

Show  from  these  relations  that 

N  =  sin  A  \/n*  -  sin1  i  —  cos  A  sin  i  (4) 

If  A  does  not  differ  from  90°  more  than  20',  a  condition 
fulfilled  in  this  case,  then  sin  A  —  1,  and  cos  A  =  0  very 
approximately,  and  the  formula  then  becomes 
N  =  \/n2  -  sin*i 

Using  this  relation  and  the  tabular  values  for  the 
refractive  index  of  water  at  the  temperature  of  your 
observation  calculate  the  refractive  index  of  the  glass 
prism. 

From  this  value  of  the  refractive  index  of  the  prism, 
and  the  values  of  the  angle  observed  above  calculate  the 
refractive  indices  of  the  other  liquids. 

Carry  five  significant  figures  throughout  all  calculations. 


97]  REFRACTION  THROUGH  A  THIN  LENS  33 

Thin  Lenses. 

A  thin  lens  is  a  lens  whose  thickness  may  be  neglected 
in  comparison  with  its  focal  length.  In  measuring  to 
the  lens,  the  student  should  measure  to  the  center  of  a 
symmetrical  lens  and  to  the  convex  surface  of  a  plano- 
convex lens. 

Direct  Measurement  of  the  Focal  Length  of  Convex  Lenses 
With  Parallel  Light. 

The  following  simple  methods  of  measuring  the  focal 
length  of  convergent  lenses  directly  with  the  aid  of 
parallel  light  will  be  found  convenient  in  checking  results 
obtained  by  other  methods. 

1.  Form  an  image  of  the  sun  upon  a  screen  with  the  lens;  the 
distance  between  the  lens  and  the  screen  when  the  image  is  most 
sharply  defined  is  the  focal  length  of  the  lens. 

2.  Mount  the  lens  on  the  optical  bench  with  a  plane  mirror 
behind  it.     Adjust  its  distance  from  the  object  screen  (a  narrow 
slit  with  a  cross  hair)  until  an  image  of  the  slit  is  formed  on  the 
screen  beside  the  slit.     The  light  between  the  lens  and  mirror  is 
then  parallel  and  the  distance  from  the  lens  to  the  screen  is  the 
focal  length  of  the  lens. 

3.  Focus  a  telescope  accurately  for  parallel  light.      Place  the 
lens  in  front  of  the  objective  of  the  telescope  and  view  through  the 
telescope  some  fixed  mark — a  narrow  slit  or  a  fine  point.     Adjust 
the  distance  between  the  lens  and  object  until  there  is  no  parallax 
in  the  field  of  the  telescope.     The  distance  from  the  lens  to  the 
object  is  the  focal  length  of  the  lens. 


97.     REFRACTION  THROUGH  A  SINGLE  THIN  LENS. 
Chromatic  and  Spherical  Aberration  and  Astigmatism. 

The  object  of  this  exercise  is  to  determine  the  constants 
(radii  of  curvature  and  refractive  index)  of  a  single  thin 
lens  and  to  study  the  phenomena  of  chromatic  and 
spherical  aberration  and  astigmatism. 


34  CHROMATIC  ABERRATION  [97 

The  optical  bench  should  be  provided  with  three  slides, 
one  to  support  the  object  screen,  one  for  the  lens  holder, 
and  a  third  for  the  image  screen.  One  or  more  cross-hairs 
across  an  aperture  in  the  large  screen  or  a  piece  of  wire 
gauze  may  serve  as  object. 

Constants  of  the  Lens. 

(a)  Place  the  object  screen  at  one  end  of  the  bench, 
slide  the  lens  somewhat  past  the  middle  of  the  bench  and, 
leaving  these  two  fixed,  move  the  image  screen  away 
from  the  lens  until  the  image  seems  to  be  in  the  same  place 
as  the  cross-hairs  in  the  eyepiece,  tested  by  seeing  that 
there  is  no  relative  motion  (absence  of  parallax)  between 
the  two  on  moving  the  eye  from  side  to  side.  Measure  the 
distance  of  the  lens  from  each  screen.  Next  move  the 
image  screen  past  the  position  of  best  definition  and  ap- 
proach it  again,  moving  the  screen  tQward  the  lens.  Meas- 
ure the  distance  from  the  image  screen  to  the  lens.  The 
average  of  the  two  values  obtained  for  the  image  distance 
v  with  the  value  of  the  object  distance  u,  constitute  one 
pair  of  values  for  the  determination  of  the  focal  length. 

Determine  a  second  pair  of  values  in  which  u  and  v 
are  about  equal. 

Take  the  necessary  measurements  with  the  spherom- 
eter  for  the  calculation  of  the  radii  of  curvature  of  the 
lens  surfaces. 

Chromatic  Aberration. 

(6)  Since  the  focal  length  of  a  lens  depends  upon  its 
index  of  refraction,  which  varies  for  different  wave-lengths, 
it  is  evident  that  the  value  of  /  determined  in  (a)  will  be 
an  average  value  for  the  brighter  portion  of  the  spectrum 
of  the  light  used.  In  order  to  determine  /  for  two  fairly 
definite  regions  of  the  spectrum,  first  put  in  front  of  the 


97]  SPHERICAL  ABERRATION  35 

object  a  piece  of  red  glass,  and  determine  a  pair  of  con- 
jugate foci.  Then  replacing  the  red  glass  with  a  piece 
•  of  blue  glass,  determine  a  second  pair  of  conjugate  foci 
for  blue  light.  As  neither  of  the  colors  are  absolutely 
pure  it  will  not  be  possible  to  eliminate  parallax  entirely. 

Spherical  Aberration. 

(c)  In  order  to  avoid  confusing  chromatic  abberation 
with   spherical   aberration,   observations   must  be   taken 
with  light  which  is  fairly  monochromatic.    This  is  best 
secured  by  using  the  red  glass  as  it  transmits  a  fairly 
narrow  region  of  the  spectrum. 

Cover  the  lens  with  a  diaphragm  having  a  one  centi- 
meter central  .opening  and  after  locating  the  image 
measure  the  conjugate  foci. 

Make  a  second  determination  of  the  image  after  the 
diaphragm  has  been  replaced  by  one  which  covers  the 
center  of  the  lens  and  lets  light  pass  only  through  the 
outer  portion. 

Astigmatism. 

(d)  With  the  red  glass  still  in  place  and  using  the 
diaphragm  with  central  opening  rotate  the  lensholder  in 
its  slide  45  degrees  about  a  vertical  axis.    This  will  make 
the  wave-front  emerging  from  the  lens  astigmatic  and  the 
definite  focus  previously  obtained  will  disappear.     Bach 
point  of  the  object  will  now  have  two  line  images  but 
no  point  image  as  before.     Since  the  object  consists  of 
vertical  and  horizontal  lines  there  will  be  one  position  in 
which  the  vertical  lines  are  in  focus  and  another  in  which 
the  horizontal  lines  are  in  focus. 


3t>  THICK  LENSES  [98 

L/ocate   both    of   these    positions   for    some    particular 
value  of  u. 


(e)  From  the  data  taken  in  (a)  and  (b)  calculate  the 
radii  of  curvature  of  the  lens  surfaces,  its  focal  length, 
and  refractive  index  for  red,  blue  and  white  light. 

Defining  the  difference  between  the  focal  length  for 
red  light  and  the  focal  length  for  blue  light  divided  by 
the  mean  focal  length  as  a  measure  of  the  chromatic 
aberration,  calculate  its  value  from  your  data. 

From  the  data  taken  in  (c)  determine  the  difference 
between  the  focal  length  for  the  center  of  the  lens  and 
for  the  edge  divided  by  the  mean  focal  length  as  a  measure 
of  the  spherical  aberration. 

Illustrate  with  diagrams,  chromatic  and  spherical 
aberration  and  astigmatism.  (See  figure  8.) 

98.     THICK  LENSES. 
Principal  Points,  Nodal  Points  and  Focal  Length. 

The  conjugate  focal  relation  l/u  +  l/v  =  I//,  in  which 
u,  v,  and  /  are  measured  .from  the  center  of  the  lens, 
holds  only  so  long  as  the  thickness  of  the  lens  may  be 
neglected  in  comparison  with  these  quantities.  For  a 
thick  lens,  it  is  possible  to  select  two  points  on  the  axis 
of  the  lens,  so  situated  that  if  the  object  distance  U  be 
measured  from  the  one  of  them,  and  the  image  distance  V 
be  measured  from  the  other,  we  obtain*  a  similar  conju- 
gate focal  relation,  namely  I/  U  +  l/V  =  l/F. 

These  two  axial  points  are  called  the  "principal" 
points  of  the  thick  lens.  Planes  drawn  perpendicular  to 

*Proof  of  this  proposition  is  given  in  Edser's  Light  for  Students, 
Chapter  VII. 


98] 


PRINCIPAL  POINTS  AND  NODAL  POINTS 


37 


the  axis  through  the  principal  points  are  called  the 
principal  planes  of  the  lens.  It  may  be  shown  that  the 
principal  planes  are  planes  of  unit  magnification.  As  a 
consequence  any  ray  directed  towards  a  point  n  in  the 
first  principal  plane,  at  a  distance  x  from  the  axis,  will 
give  rise  to  a  transmitted  ray  proceeding  from  a  point  o 
in  the  second  principal  plane,  on  the  same  side  of  the 
axis  at  a  distance  x  from  it. 

The  graphic  determination  of  images  follows  readily 

from  the  properties  of  the  principal  planes  (See  Figure  14). 

O 


Fig.    14.     Showing  the  graphic  determination  of  the  image  /  of  an  object  0  using 
the  principal  planes  PI,  Pj. 

There  are  two  other  points  on  the  axis  of  a  thick  lens 
which  possess  important  properties.  A  ray  of  light 
directed  toward  the  first  of  these  points  on  the  axis  will 
after  refraction  by  the  lens  proceed  from  the  second 
point  on  the  axis  in  a  direction  parallel  to  that  of  the 
incident  ray.  These  points  are  termed  the  first  and 
second  nodal  points  respectively.  When  image  and 
object  are  situated  in  the  same  media,  for  instance  a 
glass  lens  is  used  in  air,  the  principal  points  and  the 
nodal  points  coincide. 

The  two  principal  foci,  the  two  principal  points,  and  the 
two  nodal  points  are  the  six  cardinal  points  of  a  thick  lens. 

The  characteristic  of  the  nodal  points  may  be  made 
the  basis  of  their  determination.  If  we  have  a  ray  ac 
incident  through  the  first  nodal  point  Ni  from  a  distant 


38 


PRINCIPAL  POINTS  AND  NODAL  POINTS 


[98 


object,  it  will  emerge  parallel  through  N-2  and  cut  the 
screen  at  C.  C  will  be  the  image  of  the  point  of  the 
object  from  which  ac  came.  As  the  object  is  supposed 
to  be  at  a  great  distance  its  image  will  be  formed  in  the 
focal  plane. 

Now  suppose  the  lens  system  rotated  about  the  second 
nodal  point  N2,  as  ac  is  coming  from  a  great  distance, 
the  ray  from  the  same  point  of  the  object  aci  incident 
through  N!  will  still  be  practically  parallel  to  ac.  Thus 
the  ray  that  emerges  from  N2  has  not  moved,  and  the 
image  at  C  will  remain  at  rest.  Any  movement  of  Nz, 
however,  will  cause  C  to  move  also.  (See  Figure  15a). 


C 


Fig.  150  156  15C 

Finding  the  nodal  points  and  focal  length  of  a  lens  system 

Thus  to  find  the  second  nodal  point  N2  we  have  to 
find  the  point  about  which  rotation  of  the  lens  system 
produces  no  movement  of  the  image  of  a  distant  object. 
This  is  easily  accomplished  experimentally  because  if 
the  second  nodal  point  N2  be  in  front  of  the  axis  of  rota- 
tion 0,  i.  e.,  too  near  the  screen  (Figure  156),  then  a 
small  counter-clockwise  rotation  will  carry  N2  to  A/V 
and  therefore  C  to  C'.  If,  on  the  contrary,  the  second 
nodal  point  N2  is  behind  the  axis,  i.  e.,  too  far  from  the 
screen  (Figure  15c),  a  similar  counter-clockwise  rotation 
will  take  it  to  A/V'  and  the  image  C  moves  to  C" . 

(a)  Place  a  piano  convex  lens  in  the  nodal  slide  at 
a  distance  approximately  equal  to  the  focal  length  of 
the  lens,  and  adjust  the  plane  mirror  back  of  it  so  that 
the  light  from  the  slide  rendered  parallel  by  the  lens  will 


98]  PRINCIPAL  POINTS  AND  NODAL  POINTS  39 

be  reflected  backwards  and  form  an  image  of  the  opening 
on  the  white  screen  in  the  plane  of  the  slit  itself.  Using 
the  method  outlined  above,  locate  the  second  nodal 
point  of  the  lens  and  measure '  the  distance  from  the 
center  of  rotation  to  the  screen.  This  is  the  focal  length 
F.  Note  the  position  of  the  lens  holder  on  the  nodal  slide. 

Rotate  the  slide  180°,  and  find  the  other  nodal  point, 
measuring  the  focal  length  and  noting  again  the  position 
of  the  lens  holder  on  the  slide. 

Measure  the  thickness  of  the  lens  with  the  calipers. 

.  (b)  Determine  the  nodal  points  and  focal  length  of 
the  combination  of  a  convex  and  a  concave  lens  separated 
two  or  three  centimeters.  Measure  the  thickness  of 
each  lens  used  and  also  the  distance  apart  of  the  inner 
surface  of  the  two  lenses. 


(c)  From  the  data  taken  in  (a)  find  the  distance 
between  the  nodal  points  expressed  as  a  decimal  part  of 
the  thickness  of  the  lens.  Draw  an  enlarged  diagram 
of  the  lens,  and  on  the  axis  of  the  lens  show  the  position 
of  the  lens  surfaces  and  the  two  nodal  points. 

Show  in  a  diagram  drawn  to  scale  the  results  obtained 
in  (b). 


40  RESOLVING  POWER  OF  TELESCOPE  [99 

99.     ASTRONOMICAL  TELESCOPE. 
MAGNIFYING  POWER.     RESOLVING  POWER. 

Magnifying  Power  of  Telescope. 
Reference. — Duff,  p.  615;  Kimball,  p.  653. 

The  magnification  produced  by  a  telescope,  when 
focussed  on  a  distant  object  and  adjusted  to  suit  the 
unaccomodated  eye,  is  equal  to  the  focal  length  of  the 
objective  divided  by  the  focal  length  of  the  eyepiece. 

(a)  Remove   the    objective    and    determine   its   focal 
length  by  any   method  you  please.      (No.   2,   page  33, 
for  instance.) 

(b)  To   determine  the  focal  length   of   the   eyepiece, 
first  put  a  micrometer  eyepiece  on  the  telescope  in  place 
of  the  objective.     Using  the  eyepiece  whose  focal  length 
is  to  be  determined  as  an  objective  form   a  magnified 
image  of  a  standard  0.2  mm.  glass  scale  in  the  field  of  the 
micrometer  eyepiece.     Measure  with  the  micrometer  the 
dimensions  of  this  image. 

Now  extend  the  draw  tube  a  measured  distance  and 
again  measure  the  image  formed. 

Resolving  Power  of  Telescope. 

The  action  of  a  lens,  S,  in  forming  a  real  image,  /, 
of  a  point  source,  O,  consists  in  transforming  the  spherical 
waves,  such  as  a  b  c,  diverging  from  O  into  spherical 
waves,  such  as  d  e  f,  converging  toward  /.  From  geo- 
metrical considerations  alone  we  might  conclude  that  the 
image  at  I  is  a  point,  but  the  wave  theory  shows  that 
this  cannot  be  so. 


99] 


RESOLVING  POWER  OF  TELESCOPE 


41 


To  determine  the  effect  at  /  of  the  wave,  d  e  f,  let  us 
apply  Huygens  principle  and  consider  the  resultant 
effect  as  due  to  wavelets  with  their  centers  in  the  surface 
d  e  f.  These  wavelets  will  all  reach  /  in  the  same  phase, 
since  it  is  equidistant  from  all  of  them,  hence  7  must 
be  a  point  of  maximum  brightness.  At  points  outside 


of  I  differences  of  phase  will  exist,  and  there  must  be 
some  point,  H,  which  is  on  the  average  a  half  wave 
length  farther  off  from  the  upper  half  of  the  surface  d  e  f 
than  it  is  from  the  lower  half.  At  that  point  waves 
from  one  half  of  the  surface  will  interfere  with  those 
from  the  other  half  and  produce  complete  darkness. 
But  between  /  and  H  the  interference  is  only  partial 
and  consequently  the  light  intensity  must  fade  off  from 
J  to  H.  Since  the  light  is  symmetrical  about  e  I,  there 
must  be  a  little  spot  of  light  formed  in  the  image  plane 
having  the  distance  I H  as  its  radius.  An  optical  image 
is  thus  essentially  a  diffraction  figure.  The  height  of  the 
curve  in  Fig.  16(6),  is  proportional  to  the  intensity  of  the 
light  in  the  image  plane.  It  is  evident  that  if  we  decrease 
the  aperture  of  the  lens  the  corresponding  point  H1  will 
be  farther  from  /  than  H,  and  the  intensity  curve  will 
be  broader,  as  in  Fig.  17 (a).  And  conversely  increasing 
the  aperture  will  steepen  the  curve.  Consequently  if 
a  lens  forms  an  image  of  two  small  objects,  while  the 
central  maximum  of  each  image  is  fixed  by.  the  .distance 


42 


RESOLVING  POWER  OF  TELESCOPE 


[99 


between  the  objects  and  their  distance  from  the  lens, 
the  amount  of  overlapping  of  the  intensity  curves  will 
then  depend  upon  the  aperture  only. 

If  we  use  a  rectangular  aperture  with  the  lens,  the 
conditions  for  the  formation  of  maxima  and  minima  will 
correspond  to  those  of  a  narrow  slit.  The  "limit  of 
resolution"  is  reached  when  the  first  maximum  of  the 
image  of  one  object  coincides  with  the  first  minimum 


Fig.  17(c) 
Intensity  Curve,  Narrow  Slit 


Fig.  17(6) 
Limit  of  Resolution. 


of  the  other  as  in  Fig.  17(6).  From  this  it  may  be  shown 
that  if  d  is  the  distance  between  two  details  of  an  object 
at  a  distance  D  from  an  aperture  whose  width  parallel 
to  these  details  is  a,  they  may  be  distinguished  if 


where   X  represents  the  wave-length  of  the  light  used. 
This  ratio  is  called  the  "resolving  power"  of  the  aperture. 

(c)  Illuminate  the  metal  gauze  with  sodium  light 
and  observe  it  through  the  telescope,  which  should  have 
an  adjustable  slit  placed  just  before  its  objective. 

Adjust  the  width  of  the  slit  until  the  wires  parallel  with 
its  length  may  be  just  distinguished  and  then  measure  the 


99]  RESOLVING  POWER  OF  TELESCOPE  43 

width  of  the  slit  a  with  the  comparator.  The  distance 
D  will  be  measured  with  sufficient  accuracy  with  the  meter 
rod.  For  the  wave-length,  X,  consult  the  tables.  The 
distance  d  between  the  centers  of  the  adjacent  wires  of 
the  gauze  may  be  conveniently  determined  by  measuring 
the  distance  occupied  by  a  known  number  of  wires. 


(d)  Letting  Wi  =  Vi/Ui  be  the  magnification  of  the 
first  image  measured  in  (b),  m2  =  v2/u2  the  magnification 
of  the  second  image  and  d  =  v2-Vi  be  the  extension  of 
the  draw  tube  prove  that  the  focal  length  of  the  eyepiece 
/  =  d/(mz  -  Wi). 

Using  your  data  in  (a)  and  (6)  calculate  the  magni- 
fying power  of  the  telescope. 

Calculate  the  ratio  d/D  and  compare  it  with  the  ratio 

X/a. 

._-— • 

Explain,  using  a  diagram,  the  formula  given  above  for 
the  resolving  power. 


44  THE  SPECTROSCOPE  [100 

100.     THE  SPECTROSCOPE. 
Drawing  Spectra.     Calibration  of  Spectroscope. 

The  direct  vision  spectroscope  consist  of  two  tubes,  H, 
r,  the  first  contains  a  slit  5,  adjustable  by  means  of  a 

H 


\L 


Fig.  18.     Direct  vision  spectroscope. 

knurled  ring  K,  and  protected  by  means  of  a  glass  disk  g, 
placed  just  in  front  of  it;  the  second  tube  movable  within 
the  first,  carries  a  short  focus  lens  L,  and  the  direct  vision 
prism  P.  When  adjusted  for  the  unaccomodated  eye  the 
lens  renders  the  light  from  the  slit  parallel  before  it  enters 
the  prism. 

A  small  tube,  attached  at  right  angles  to  the  axis  of  the 
movable  tube  contains  a  scale  s,  a  lens  system  /,  which 
may  be  moved  along  the  tube,  and  an  adjustable  mirror  M, 
for  illuminating  the  scale.  The  light  from  the  scale,  having 
been  rendered  parallel  by  the  lens,  is  reflected  on  the 
slanting  end  of  the  prism  and  reaches  the  eye  at  E,  with 
the  light  which  has  gone  through  the  prism.  To  adjust 
the  spectroscope,  the  lens  system  /,  in  front  of  the  scale, 
should  be  moved  by  means  of  the  pin  d,  until  the  scale  is 
clearly  seen  and  then  the  tube  containing  the  prism  should 
be  moved  in  or  out  until  the  spectrum  is  sharply  defined 
and  is  seen,  without  parallax,  partly  overlapping  the 
image  of  the  scale. 


100]  CALIBRATION  OF  SPECTROSCOPE  45 

The  position  of  lines  of  a  particular  wave-length  on  the 
spectroscope  scale  depends  not  only  upon  the  scale, 
which  is  arbitrary,  but  also  upon  the  dispersion  of  the 
glass  prism  used.  Scale  readings  of  a  spectroscope  must 
accordingly  be  put  into  wave-lengths  for  purposes  of 
comparison  with  the  values  obtained  by  other  methods. 
Data  for  a  calibration  curve  consists  of  a  series  of  scale 
readings  for  lines  of  known  wave-length.  These  should 
be  distributed  with  some  uniformity  over  the  spectrum- 
Suitable  lines  may  be  found  in  the  spectrum  of  sun- 
light— the  "Fraunhofer  lines" — or  in  the  flame  spectra 
of  the  light  metals.  If  both  sources  are  available  the 
calibration  curve  obtained  using  one  may  be  used  to 
determine  the  wave-length  of  the  lines  in  the  spectrum 
of  the  other. 

To  examine  the  flame  spectra  of  the  light  metals. 

(a)  Color  the  Bunsen  flame  with  the  salts  furnished 
and  examine  their  spectra. 

Note  in  each  case  the  general  color  of  the  flame  and 
the  colors  of  the  various  bands  and  lines. 

To  observe  successfully  the  potassium  spectrum  it  will 
be  necessary  to  open  the  slit  somewhat  and  insert  a  piece 
of  cobalt  glass  between  the  flame  and  slit.  Remove  the 
light  illuminating  the  slit  until  the  line  is  seen,  then 
replace  the  light  and  take  the  scale  reading. 

The  sodium  spectrum  will  probably  be  ever  present, 
but  it  is  readily  distinguished  from  that  of  the  salt  under 
examination.  Protect  the  eye  from  side  light,  especially 
when  locating  the  fainter  lines. 

To  examine  the  solar  spectrum  with  the  spectroscope. 

(6)  Reflect  sunlight  so  that  it  falls  on  the  slit,  nar- 
rowing the  latter  to  the  width  of  a  fine  hair.  If  the 


46  ABSORPTION  SPECTRA  [100 

4aBC      D          Eb    c  F      d         eGg        h    HK 


Red,  yellow,  green,  blue.  indigo,  violet. 

Fig.    19.     The  Fraunhofer  lines. 

instrument  has  been  properly  focused  a  continuous 
spectrum  crossed  by  fine  vertical  dark  lines — "Fraun- 
hofer's  lines"  will  be  visible. 

Note  the  position  and  color  of  the  principal  lines, 
identify  them  with  the  aid  of  the  figure.  Some  direct 
vision  prisms  have  a  smaller  dispersion  in  the  red  and 
much  larger  dispersion  in  the  extreme  violet  than  here 
indicated. 

Absorption  Spectra. 

(c)  Draw  the  spectrum  of  a  luminous  flame  seen 
through  red,  yellow,  green,  and  blue  glass. 


(d)  Plot  a  curve  with  the  scale  readings  taken  in  (6) 
as  ordinates,  and  the  corresponding  wave-lengths  given 
in  the  tables  as  abscissae.  From  this  plot  determine  the 
wave-length  of  the  sharpest  lines  observed  in  (a). 

Draw  the  spectra,  observed  in  (c),  upon  plotting  paper, 
the  spectroscope  scale  being  taken  as  abscissae  and  each 
spectrum  occupying  a  separate  horizontal  strip  about  a 
centimeter  high. 


101]  SPARK  SPECTRA  OF  METALS  47 

101.     SPARK  SPECTRA  OF  METALS. 

The  spectroscope  which  has  previously  been  adjusted 
and  calibrated  should  be  used  for  this  exercise. 

The  spark  spectra  of  various  metals  are  to  be  observed 
and  the  wave-length  of  the  lines  to  be  determined  from 
their  position  on  the  scale  as  calibrated. 

(a)  Connect   the   two   electrodes   of   the   metal   with 
the  secondary  of  the  induction  coil.    The  storage  battery 
(6-8  volts)   should  be  connected  to  the  primary  of  the 
coil.      The   spark   gap   should   be   horizontal   and   about 
1   millimeter  in  width.     Start  the  coil  and  observe  the 
spark  directly.     Describe  the  change  in  the  appearance 
of  the  spark  when  a  Ley  den  jar  is  connected  in  parallel 
with  the  secondary  of  the  coil. 

(b)  Leave  the  Ley  den  jar  connected  as  in  (a).     Set 
the  spark  gap  in  front  of  the  slit  of  the  spectroscope  and 
adjust  the  height  of  the  spark  so  that  the  spectrum  is 
seen  in  the  telescope  above  the  illuminated  scale.     Locate 
the  lines  on  the  scale,  tabulating  the  readings  as  follows: 
metal,  color,  scale  reading  and  wave-length,  as  read  from 
the  calibration  curve.     After  color  place  a  figure  1,  2  or 
3,  to  designate  the  intensity.    The  brightest  lines  being  1. 

(c)  Introduce  self-induction  into  the  circuit  by  con- 
necting the  coil  furnished  in  series  with  the  spark  gap. 
Note  the  change  in  the  spectrum.     By  comparing  the 
present  spectrum  with  the  readings  of   (b)   make  a  list 
of  the  lines  which  ha,ve  entirely   disappeared.     A  con- 
venient method  of  making  the  comparison  is  to  use  a 
commutator  irt  making  the  connections,  then  by  reversing 
the   commutator   the   self-induction   may   be   thrown   in 
or   out. 

(d)  Repeat  (b)  and(c)  with  each  of  the  other  metals 
furnished.     To  what  substance  do  the  lines  belong  which 
diasppear  when  self-induction  is  put  into  the  circuit? 


48 


NEWTON'S  RINGS 


[102 


102.     WAVE-LENGTH  OF  SODIUM  LIGHT. 
Newton's  Rings. 

If  a  plano-convex  lens  be  placed  upon  a  plane  piece  of 
glass,  a  thin  air  film  will  be  formed  between  them.  Near 
the  point  of  contact  the  thickness  of  the  film  will  be  small 
compared  with  the  wave-length  of  light,  consequently 
there  will  be  a  circular  black  spot  around  the  point  of 
contact  when  this  is  viewed  by  reflected  light.  The  air 
film  increases  in  thickness  from  the  point  of  contact 
outwards  in  every  direction,  and,  since  the  lower  surface 
of  the  lens  is  spherical,  points  of  equal  thickness  form 
concentric  circles  with  the  point  of  contact  as  the  center. 

Using  monochromatic  light  to  illuminate  the  lens  and 
plate,  the  central  black  spot  will  be  seen,  by  reflected 
light,  surrounded  by  concentric  bright  circles  separated 
by  dark  intervals. 

Interference*  has  taken 
place  between  the  light  re- 
flected from  the  lower  sur- 
face of  the  lens  and  that 
reflected  from  the  glass  plate, 
the  optical  path  of  the  latter 
being  greater,  by  twice  the 
thickness  of  the  air  film, 
than  that  of  the  first. 


(a)     Use  sodium  light  as 
a  source,  placing  the  burner 
at  the  focus  of  the  lens.    The 
plate  glass  (G)  should  be  set 
so  as  to  reflect  the  light  downward.     (See  fig.)  If  a  broad 


Fig.  18. 


Arrangement  of  the  apparatus 
for  observing  Newton's  Rings. 


"This  is  a  pure  interference  phenomenon,  the  difference  of  phase 
necessary  for  the  formation  of  the  rings  being  introduced  without 
violation  of  any  of  the  laws  of  geometrical  optics. 


102]  NEWTON'S  RINGS  49 

flame  is  used  as  a  source,  the  lens  L,  may  be  dispensed 
with.  Remove  the  microscope  from  its  holder  and  focus 
the  eyepiece  on  the  cross-hairs.  Adjust  the  plate  glass 
(G)  so  that  the  light  reflected  downwards,  returns  to  the 
eye  after  reflection  on  the  lens  and  plate  at  (P).  Find 
the  rings,  move  the  microscope  holder  to  get  them 
roughly  in  the  center  of  the  opening,  and  then  insert  the 
microscope,  focusing  carefully  with  the  aid  of  the  cross- 
hairs. In  the  proper  position  there  will  be  no  relative 
motion  of  the  rings  with  reference  to  the  cross-hairs  on 
moving  the  source  of  light. 

Take  care  to  prevent  lost  motion  (backlash)  by  always 
turning  the  screw  in  the  same  direction  when  making  a 
setting. 

Measure  the  diameter  of  several  rings  near  the  cen- 
ter, then  skip  to  the  tenth,  or  beyond,  and  measure  sev- 
eral more,  finding  the  difference  in  diameter  as  accurately 
as  possible  by  repeated  measurement  of  successive  rings 
on  one  side,  then  move  to  the  opposite  side  and  repeat 
the  measurements.  The  lens  should  not  be  disturbed 
during  the  entire  series  of  observations. 


(b)  Show  with  the  aid  of  a  drawing  that  if  r  is  the 
radius  of  curvature  of  the  lens  surface,  Di,  D2,  D3,  D4, 
Dn,  the  diameters  of  successive  bright  rings,  counting 
from  the  center  and  X  the  wave-length  of  the  light  used 


2r(2n-l] 

For  the  (n-f  AOth  ring,  X  =  -  (2) 


Solving    for  the  numerators  in  (1)  and  (2)  and  taking 
their  difference  we  have 


50  THE  GRATING  CONSTANT  [102-103 

This  final  form  gives  us  an  expression  for  the  wave- 
length which  is  independent  of  the  actual  thickness  of 
the  air  film  at  the  point  of  contact.  It  also  shows  that  the 
results  will  be  mare  accurate  the  farther  apart  the  rings 
are  whose  diameters  are  used. 

In  what  two  ways  is  the  difference  of  phase  between 
the  interfering  rays  produced? 

(c)  Suggest  a  modification  of  the  experiment  by  which 
the  center  would  appear  bright  instead  of  dark  by  reflected 
light. 

(d)  From  the  known  value  of  r  and  your  data,  calcu- 
late the  value  of  the  wave-length  of  sodium  light. 

Find  the  mean  error  in  per  cent  in  your  result. 

Using  the  average  value  for  the  wave-length  obtained 
in  (d)  calculate  the  actual  thickness  of  the  air  film  having 
the  same  diameter  as  the  third  bright  ring,  assuming  the 
lens  and  plate  in  actual  contact. 

103.     THE  DIFFRACTION  GRATING. 

The  Grating  Constant. 
Reference — Edser,  Light  for  Students,  p.  452. 

A  monochromatic  light  is  required.  For  this  purpose 
use  a  Bunsen  burner  specially  arranged  to  give  a  strong 
flat  sodium  flame.  Set  in  front  of  this  the  screen  with  its 
plane  vertical  and  the  slits  horizontal.  In  use,  the  light 
is  allowed  to  shine  through  one  slit  on  each  side  of  the 
central  vertical  line  of  the  screen. 

The  grating  is  mounted  in  a  slide  on  the  optical  bench, 
with  the  ruled  surface  toward  the  screen  and  the  lines 
parallel  to  the  slits.  On  looking  at  the  slits  through  the 
grating,  a  series  of  diffracted  images  of  each  slit  is  seen. 
The  two  sets  of  images  move  relatively  to  each  other  as 


103] 


THE  GRATING  CONSTANT 


51 


the  grating  is  moved  back  and  forth.  A  position  of  the 
grating  may  be  found  for  which  one  of  the  images  in  one 
set  appears  to  form  a  continuous  line  with  one  in  the 
other  set.  If  the  lines  do  not  quite  meet,  it  is  evident  that 
the  lines  of  the  grating  are  not  quite  parallel  to  the  slits. 

(a)  Using  the  two  slits  which  are  closer  together,  set 
the  grating  so  that  the  first  image  of  one  slit  coincides  in 
the  way  above  indicated,  with  the  first  image  of  the  other 
slit,  at  the  point  half-way  between  the  two  slits.    Make  a 
number  of  settings  and  for  the  mean  of  these  settings 
determine  the  distance  of  the  ruled  surface  of  the  grating 
from  the  screen. 

(b)  Repeat  (a),  using  the  second  diffracted  image  of 
each  slit,  instead  of  the  first. 

(c)  Repeat  (a)  and  (6)  for  the  pair  of  slits  which  are 
wider  apart. 

(d)  Measure  the  distance  between  the  slits.     Deter- 
mine the  mean  distance  from  measurements  on  the  edges 
taking  a  number  of  readings  for  each  distance. 


(e)  Explain  the  formation  of  both  the  second  and 
fifth  diffracted  images  in  the  final  position  .secured  in  (b) 
with  the  aid  of  a  drawing,  lettered  like  the  figure  given. 
Are  the  images  real  or  virtual?  Derive  an  expression  for 


Fig.    19.      Determination  of  the  Grating  Constant  using  two  slits. 


52  MEASUREMENT  OF  WAVE-LENGTH        [103-104 

the  distance  apart  of  the  lines  of  the  grating — the  grat- 
ing constant,  in  terms  of  the  wave-length  of  the  light 
used,  the  distance  from  the  screen  to  the  grating  and  the 
distance  between  the  slits. 

(/)  Assuming  the  wave-length  of  sodium  light  to  be 
0.00005893  cm.,  calculate  the  grating  constant. 

104.     MEASUREMENT  OF  WAVE-LENGTH. 
With  Spectrometer  and  Grating. 

The  telescope  and  collimator  should  be  focused  for 
parallel  light.  (See  directions  Bxp.  94.) 

Mount  the  grating  on  the  table  of  the  spectrometer, 
with  the  lines  vertical. 

To  set  the  grating  perpendicular  to  the  collimator 
clamp  the  telescope  at  an  angle  of  90°  with  the  collimator, 
then,  loosening  the  table  rotate  it  until  the  slit  is  seen 
by  light  reflected  from  the  face  of  the  grating.  Noting 
this  position,  move  the  table  through  45°  and  the  grating 
will  be  perpendicular  to  the  collimator. 

In  case  the  mounting  of  the  grating  does  not  permit 
the  use  of  the  'above  method,  find,  by  trial,  a  position  of 
the  grating  such  that  the  deviation  for  the  spectrum  of 
the  first  order  is  the  same  on  both  sides,  within  half  a 
degree.  This  may  be  done  without  reading  the  vernier, 
the  accurate  determinations  of  the  angle  being  made  later. 

(ai)  Work  this  or  the  following  section.  Determine 
the  wave-length  of  the  principal  lines  in  the  spectrum* 
of  the  salts  furnished.  Take  observations  both  to  the 
right  and  the  left  for  as  many  orders  as  permit  of  accurate 
measurements.  Arrange  the  data  in  tabular  form,  giving 

*The  most  satisfactory  method  of  securing  a  spectrum  of  great 
intensity  is  to  use  a  carbon  arc,  fed  with  salts,  a  separate  pair  of 
carbons  being  used  for  each  salt. 


104-105]  REFLECTION  ON  TRANSPARENT  SUBSTANCES  53 

for  each  metal  the  color  of  the  lines  and  the  average 
values  for  the  wave-length. 

(a2)  Determine  the  wave-length  of  the  two  "£)"  lines 
of  the  sodium  flame. 

(b)  Show  with  the  aid  of  a  diagram,  the  relation  which 
exists  between  the  wave-length  of  the  light  used,  the 
grating  constant,  and  the  angular  distance  of  the  spectra 
of  the  first,  second,  and  third  orders,  measured  from 
the  image  of  the  slit  seen  direct. 

The  constant  of  the  grating  used  will  be  furnished. 

105.     REFLECTION  OF  LIGHT  ON  TRANSPARENT 
SUBSTANCES. 

As  a  source  of  light  use  the  flame  of  a  candle,  or  better, 
a  sodium  flame,  if  available,  against  a  dark  background. 
The  reflecting  surface  is  glass,  the  back  surface  of  which 
has  been  blackened;  an  over-exposed  photographic  plate 
works  well  when  black  glass  is  not  to  be  had. 

I.  To  examine  the  light  coming  direct  from  the  flame. 

(a)  Place  the  flame  between  the  two  reflectors  with 
the  flame  at  the  same  height  as  the  axis  of  rotation  of 
the  plate.      Find  the  image  of  the  flame  in  the  glass; 
rotate  the  glass  about  the  incident  light  as  an  axis.     Do 
you  note  any  change    in    the    intensity    of  the  flame? 
Change  the  angle  of  incidence  by  shifting  the  flame.     If 
you  detect  any  change  in  the  intensity  of  the  flame  as 
the  reflecting  surface  is  rotated  to  what  may  it  be  due? 

Diagram  the  apparatus  as  arranged  and  make  a  state- 
ment covering  your  observation. 

II.  To  examine  the  light  from  the  flame  after  one  re- 
flection on  glass. 

(b)  Use    two    reflecting    surfaces    having   the   second 
movable  about  the  incident  ray.     Place  the  flame  so  that 


54      REFLECTION  ON  TRANSPARENT  SUBSTANCES    [105 

it  may  be  seen  in  surface  2,  the  light  being  first  reflected 
on  surface  1.     Examine  the  image  as  surface  2  is  rotated. 

Diagram  the  apparatus  as  now  arranged. 

In  case  you  observe  no  appreciable  change  in  either 
(a)  or  (6)  repeat  both  with  different  angles  of  incidence. 

If  you  note  a  change  in  the  intensity  of  the  flame,  try 
to  increase  this  change  by  varying  successively  the  angle 
of  incidence  of  the  light  on  the  first  surface  and  then  by 
rotating  the  second  surface.  In  case  repeated  trials  fail 
to  show  any  difference,  rest  the  eyes  for  a  few  minutes 
and  avoid  looking  at  any  bright  light  while  making  a 
new  trial;  do  not  look  at  the  flame  except  by  reflection. 
Ask  for  assistance  in  case  your  efforts  are  not  successful. 
If  you  find  that  light  once  reflected  behaves  differently 
from  light  coming  directly  from  the  source,  make  the 
two  planes  of  incidence  coincide  and  describe  the  change 
in  the  appearance  of  the  flame  as  the  second  surface  is 
rotated  from  this  initial  position. 

(c)  Diagram  the  apparatus  as  arranged  to  show  a 
maximum  of  the  observed  effect,  indicating  the  planes  of 
incidence  of  the  light  on  the  two  'surfaces.  Formulate 
the  results  of  your  observation  so  as  to  cover  the  follow- 
ing points. 

1.  This  phenomenon  has  been  called  the  "polarization" 
of  light  by  reflection.     From  this  experiment  alone  what 
meaning  would  be  attached  to  the  word  "polarized"  in 
the  phrase  "the  light  reflected  from  the  first  piece  of  glass 
is    polarized?"      To    what    conception    of    the   effect    of 
reflection  upon  the  incident  light  do  we  owe  this  word? 

2.  How  many  times  does  the  change  observed  occur 
in  one  complete  revolution  of  the  second  plate? 

3.  State  a  position  of  surface  2  with  reference  to  sur- 
face 1,  such  that  turning  an  equal  number  of  degrees  to 


105] 


BREWSTER'S  LAW 


55 


the  right  or  to  the  left  produces  the  same  effect.     The 
beam  is  evidently  symmetrical  about  such  a  position. 

Planes  of  symmetry  are  often  taken  as  reference  planes. 
Find  a  second  plane  of  symmetry  stating  the  angle  which 
it  makes  with  the  first.  The  plane  of  symmetry  (with 
reference  to  the  changes  in  intensity}  in  which  the  light  is 
most  plentifully  reflected  is  called  the  plane  of  polarization. 

4.  Would  the  phenomenon  observed  be  possible  if 
light  were  wholly  a  longitudinal  disturbance  of  the  ether? 

How  much  does  this  experiment 
tell  you  about  the  direction  of  the 
displacement  in  the  wave  front? 

Suggestions  for  Further  Work. 

Brewster's   Law.      Set  the  two 

plates  for  complete  extinction  of  the 
light  and  then  with  a  couple  of 
meter  rods  determine  the  angle  of 
incidence  for  the  light  striking  the 
first  plate  of  glass. 

State  the  relation  which  Brewster 
showed  exists  between  the  angle  of 
incidence  and  the  refractive  index 
of  the  substance. 

Nicol's  Polarizing  Prism.     Ex- 
amine the  model  of  a  Nicol  prism  and 
describe    its     construction     briefly, 
using  a  diagram. 

Determine  the  .plane  of  polariza- 
tion of  a  Nicol  prism  mounted  in  a 
brass  tube,  stating  your  result  with 
reference  to  the  angles  or  diagonals 
on  the  end  face. 


Scctfon  of  Nicol's  Prism 

Figs.  20.  21.  Nicol's  Prism 


56  PHYSICAL  TABLES 


DENSITIES  OF  SOLIDS. 

Solid  Density  Solid  Density 

gms.  per  cc.  gms.  per  cc. 

Aluminum 2.70  Iron,  cast 7.4 

Brass,  cast 8.44  Iron,  wrought... 7.8 

Brass,  drawn 8.70  Lead 11.3 

Copper 8.92  Nickel._ 8.90 

German-silver 8.62  Platinum 21.50 

Glass,  crown 2.6  Rubber,  hard... 1.10-1.27 

Glass,  flint 3.9  Silver 10.53 

Gold 19.3  Steel ..  7.8 


STANDARDS  OF  PITCH. 

The  French  Standard,  "Diapason  Normal"  of  1859  (which  adopts 
a  fork  having  c"=522  at  20°  C.)  is -coming  into  general  adoption 
for  organs  and  pianos  in  England,  the  Continent,  and  America,  as 
the  result  of  a  makers'  conference  in  1899.  Other  scales  in  vogue 
are  Concert  Pitch  (c"=546),  Society  of  Arts  (c"=528),  Tonic 
Sol-fa  (c"=507),  Philharmonic  (c"=540).  (The  "middle"  c  of  the 
piano  is  c'). 


ELASTIC  CONSTANTS  OF  SOLIDS. 
(Approximate  Values.) 

In  C.  G.  S.  Units. 

Substance                   Bulk-                   Simple  Young's 

Modulus               Rigidity  Modulus 

Aluminum                   5.5  x  10"             2.5  x  1011  .        6.5  x  1011 

Brass,  drawn            10.8  x    "                3.7  x    "  10.8  x    " 

Copper                        16.8  x    "                4.5  x    "  12.3  x    " 

German  Silver          4.5  x    "  12.8  x    " 

Glass                          2.4  x    "  7.0  x    " 

Iron,  wrought           14.6  x    "                7.7  x    "  19.6  x    " 

Steel                           18.4  x   "                8.2  x    "  21.4  x    " 


PHYSICAL  TABLES  57 


VELOCITY  OF  SOUND  IN  METERS  PER  SECOND. 

Aluminum 5100 

Brass 3200  to  3600 

Copper....  ....3500  to  3900 

Glass, ...5000  to  5900 

Iron..... .5000  to  5100 

Nickel 4970 

Steel....  ....4900  to  5000 


Gases  at  0°  C. 

Air 332 

Carbon  dioxide.., 261 

Chlorine 206 

Illuminating  Gas 490 

Hydrogen....  1285 

Oxygen....                                                                                                 ..  317 


PRINCIPAL  LINES  IN  SPARK-SPECTRA  OF 

Cadmium       Zinc  Tin  Lead  Air  Copper 

3982  4680  4525  4058  4631  4023 

4413  4722  5563  4247  5003  4063 

4678  4811  5589  4387  5006  4275 

4800  4912  5799  5373  5679  4378 

5059  4925  6453  5608  5933  4481 

5338  6103  6657  5943  4587 

5379  6362  6563  5106 

6438  5153 

5218 
5700 
5782 

The ;  colors  of  the  spectrum  are  approximately:  violet  3600  to 
4240,  blue  4240  to  4920;  green  4920  to  5350;  yellow  5350  to  5860; 
orange  5860  to  6470;  red  6470  to  8100. 


58  PHYSICAL  TABLES 

INDEX   OF  REFRACTION  OF  VARIOUS  SUBSTANCES. 

For  Sodium  Light  D  Line,  X  =  5893xlO-8  cm. 

Air,  Dry  0° . ....1.0002945 

"  20° 1.0002773 

Alcohol 1.3616 

Benzene 1.5005 

Calcite,  ordinary  ray 1.6585 

"       extraordinary  ray._ 1.4864 

Canada  Balsam,  hard 1.54 

Carbon  Bisulphide,  20° 1.62772 

Cassia  Oil,  17.5° 1.6053 

"      "     Light... 1.5153 

Glass,  crown 1.465  to  1.6112 

"      flint 1.515  to  1.75 

densest  flint '. 1.9626 

Glycerine 1.47 

Cornea,  aqueous  and  vitreous  humors  of  eye. 1.3365 

Crystalline  lens 1.4371 

a — monobromnaphthelene  15° ^ 1.6603 

20° _.. 1.6581 

Methyl  Iodide,  15°._ 1.7429 

"       20° 1.7419 

Quartz,  ordinary  ray 1.5442 

"        extraordinary  ray.... 1.5533 

Rock  Salt 1.5443 

Olive  Oil- 1.47 

Flourspar 1.43085 

Diamond : 2.4173 

Water,  15° 1.33339 

"        20° _„,. 1.33300 

22° 1.33280 

"        25° 1.33253 

30° 1.33194 

Ice,  ordinary  ray 1.3087 

"    extraordinary  ray 1.3119 

Turpentine 1.47 


PHYSICAL  TABLES  59 

WAVE-LENGTHS  OF  THE  PRINCIPAL  FRAUNHOFFER 
LINES   IN   AENGSTROEM'S   UNITS     (ten   millionth   mm.) 
(In  air  at  20°C  and  760  mm.) 


Line 


Element    Line 


K 3934 Ca 

H 3968 Ca 

h 4102 H,  In 

g 4227 Ca 

G 4308 Ca,  Fe 

e 4384 Fe 

d. 4646 Fe 

F....     ....4862....   ...H 


Element 


.5173. 


-Mg 


bt 5184 Mg 

E. 5270 Fe 

D2._ 5889.965....Na 

D!.. 5895.932....Na 

D. average 5893 Na 

C. 6563 H 

B. 6870 O 

a 7202 Atm 

A._.      ....7608 O 


LINES  OF  THE  FLAME-SPECTRA  OF  SEVERAL  METALS. 

In  the  following  table  *  denotes  an  average  for  lines  near  to- 
gether, or  the  center  of  a  band;  e  the  edge  of  a  band  nearest  the 
D  line;  m  that  the  line  belongs  to  the  metal  itself. 

The  lines  most  useful  in  identifying  the  substance  are  printed 
in  bold  face  type;  those  suitable  for  the  calibration  of  a  spectro- 
scope scale  are  marked  c. 


K 

Na 

4046mc 

5893  *mc 

Faint  con- 

tinuous 

Li 

spectrum 

6102m 

in  the 

6708mc 

blue  and 

green 

Tl 

5351mc 

7680  *mc 


Ca 

4227mc 

5530* 

5728* 

5817 

5933 

6055* 

6191 

6265* 

6441 


Sr 

4607mc 
6045* 

6233* 
6350* 
6464* 
6597* 
6694* 
6827* 


Ba 

4873* 

5089e 

5215e 

5346e 

5492e 

5536m 

5661* 

5719* 

5881* 

6044* 

6297* 


Cs 

4560mc 

4597mc 

6007m 

6219m 

Rb 

4202mc 
4216mc 
6207m 
6297m 
7811  me 


*Spectres  Lumineux,  Lecoq  de  Boisbaudran. 


60 


NATURAL  SINES. 


0' 

6' 

12' 

18 

24 

30' 

36 

42 

48' 

54' 

123 

4      5 

0° 

0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12      15 

1 

2 
3 

0175 

0349 

0523 

0192 
0366 
0541 

0209 
0384 
0558 

0227 
0401 
0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
0663 

0332 
0506 
0680 

369 
369 
369 

12      15 

12       15 
12      15 

4 
5 
6 

0698 

0872 
1045 

071507320750 
08890906  0924 
1063  1  080,1097 

0767  0785 
0941  0958 
ni5|ii32 

0802 
0976 
1149 

*323 
1495 
1668 

0819 

0993 
1167 

0837 
ion 
1184 

0854 
1028 

1  201 

369 
369 
369 

12       15 
12      14 
12      J4 

7 
8 
9 

1219 

1392 
1564 

1236 
1409 
1582 

1253 
1426 

1599 

1271 
1444 
1616 

1788 

1288 
1461 
1633 

1305 
1478 
1650 

[340 
1513 
1685 

1357 
1530 
1702 

1374 
1547 
1719 

369 
369 
369 

12      14 
12      14 
12      14 

10 

1736 

1754 

1771 

1805 

1822 

1840 

i857|i874 

I89I 

369 

12      14 

11 
12 
13 

1908 
2079 
2250 

1925 
2096 
2267 

1942 
2113 

2284 

r959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

2OII 

2181 

2351 

2028 
2198 
2368 

2045 
2215 

2385 

2062 
2232 
24O2 

369 
369 
368 

II       14 
II       14 
II       14 

14 
15 
16 

24*9 

2588 

2756 

2436 
2605 
2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 
2857 

2538 
2706 
2874 

2554 
2723 
2890 

2571 
2740 
2907 

368 
368 
368 

II       14 
II       14 
II       14 

17 
18 
19 

2924 
3090 
3256 

2940 
3107 
3272 

2957 
3123 
3289 

2974 
3140 

3305 

2990 

3156 
3322 

3007 
3173 
3338 

3024 
3190 

3355 

3040 
3206 
3371 

3057 
3223 
3387 

3074 
3239 
3404 

368 

368 

3  5  8 

II       14 
II       14 
II       14 

20 

3420 

3437 

3453 

3469 

3486 

3502 

35i8 

3535 

355i 

3567 

3  5  8 

II       14 

21 
22 
23 

3584 
3746 
39°7 

3600 
3762 
3923 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 
3971 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

3714 
3875 
4035 

3730 
3891 
4051 

3  5  8 
3  5  8 
3  5  8 

II       14 
II       14 
II       14 

24 
25 
26 

4067 
4226 

4384 

4083 
4242 
4399 

4099 
4258 
4415 

4H5 
4274 

4431 

4i3i 
4289 
4446 

4M7 
4305 
4462 

4163 
4321 

4478 

4179 

4337 
4493 

4195 
4352 
4509 

4210 
4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II     13 
II     13 
10     13 

27 
28 

29 

4540 
4695 

4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
4741 
4894 

4602 
4756 
4909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
4970 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10    13 
10    13 
10    13 

30 

5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

3  5  8 

10    13 

31 
32 
33 

5150 
5299 
5446 

5165 
53M 
5461 

5180 
5329 
5476 

5195 
5344 
5490 

5210 
5358 
5505 

5225 
5373 
5519 

5240 

5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 
5577 

257 

2  5  7 
257 

10      12 
10      12 

IO      12 

34 
35 
36 

5592 
5736 

5878 

5606 
5750 
5892 

5621 
5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

5707 
5850 
5990 

5721 
5864 
6004 

257 
257 
2  5  7 

10      12 
10      12 

9     12 

37 
38 
39 

6018 

6157 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 
6347 

6088 
6225 
6361 

6101 
6239 
6374 

6115 
6252 
6388 

61296143 
6266  6280 
6401  6414 

257 
2  5  7 
247 

9     12 
9    ii 
9     ii 

40 

6428 

6441 

6455 

6468 

6481 

6494 

6508 

652165346547 

247 

9     " 

41 
42 
43 

6561 
6691 
6820 

6574 
6704 
6833 

6587 
6717 
6845 

6600 
6730 

6858 

6613 

6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

6652 
6782 
6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
246 
246 

9     ii 
9     ii 
8     ii 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

246 

8     10 

NATURAL  SINES. 


61 


45° 

46" 
47 
48 

0' 

6' 

12' 

18 

24' 

30 

36 

42' 

48 

54 

123 

4      5 

7071 

7083 

7096 

7108 

7120 

7133 

7M5 

7157 

7169 

7181 

2    4    6 

8     10 

7i93 
7314 
743i 

7547 
7660 

777i 

7206 
7325- 
7443 

7218 
7337 
7455 

7230 

7349 
7466 

7242 
736i 
7478 

7254 
7373 
7490 

7266 

7385 
75oi 

7278 
7396 
7513 

7290 
7408 
7524 

7302 
7420 
7536 

246 
246 
246 

8     10 
8     10 
8     10 

49 
50 
51 

7558 
7672 
7782 

7570 
7683 
7793 

758i 
7694 
7804 
7912 
8018 
8121 

7593 
7705 
78_L5 
7923 
8028 
8131 

7604 
7716 
7826 

7615 
7727 
7837 

7627 
7738 
7848 

7638 
7749 
7859 

7649 
7760 
7869 

246 
246 
245 

8      9 
7      9 
7      9 

52 
53 
54 

7880 

7986 
8090 

7891 
7997 
8100 

7902 
8007 
8111 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

245 
235 
2    3     5 

7      9 
7      9 

7      8 

55 

8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

235 

7      8 

56 
57 
58 

8290 

8387 
8480 

8300 
8396 
8490 

8310 
8406 
8499 
8590 
8678 
8763 
8846 
8926 
90Q3 
9078 

8320 
8415 
8508 

8329 
8425 
8517 

8339 
8434 
8526 

8348 
8443 
8536 

8358 
8453 

8545 

8368 
8462 
8554 

8377 
8471 

8563 

235 
235 
235 

6       8 
6      8 
6      8 

59 
60 
61 

8572 
8.660 
8746 

8581 
8669 
8755 
8838 
8918 
8996 

8599 
8686 
877i 

8607 
8695 
8780 

8616 
8704 
8788 

8625 
8712 
8796 

8634 
8721 
8805 

8643 
8729 
8813 

8652 
8738 
8821 

3    4 

3     4 
3    4 

6       7 

6       7 

6       7 

62 

63 
64 

8829 
8910 
8988 

8854 
8934 
9011 

8862 
8942 
9018 

8870 
8949 
9026 

8878 
8957 
9033 

8886 
8965 
9041 

8894 
8973 
9048 

8902 
8980 
9056 

3    4 

3     4 
3    4 

5       7 
5      6 
5       6 

65 
"66" 
67 
68 
~69~ 
70 
71 
72 
73 
74 
75" 
76 
77 
78 

9063 

9070 

9143 
9212 
9278 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

2    4 

5      6 

9135 
9205 
9272 

-9150 
9219 
9285 

9157 
9225 
9291 

9164 
9232 
9298 

9361 
9421 
9478 
9532 
9583 
9632 

9171 
9239 
9304 

9367 
9426 

9483 

9537 
9588 
9636 

9178 
9245 
93ii 

9184 
9252 
9317 

9191 
9259 
9323 

9i9h 
9265 
9330 

2     3 
2     3 
2     3 

5      <> 
4      6 

4       5 

9336 
9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
9415 
9472 

9527 
9578 
9627 

9373 
9432 

9489 

9379 
9438 
9494 

9385 
9444 

939  T 
9449 
9505 

2     3 
2     3 
2    3 

4       5 
4       5 
4       5 

9500 

95ii 
9563 
9613 

9659 
9703 
9744 
9781 

95i6 
9568 
9617 

9521 
9573 
9622 

9542 
9593 
9641 

;68G 

9548 
9598 
9646 

9553 
9603 
9650 

955« 
9608 
9655 

2     3 
2      2 
2      2 

4      4 
3      4 
3      4 

9664 

9668 

9673 

9677 

9681 

9690 

9694 

,699 

I      2 

3      4 

9707 
9748 
9785 

9711 

9751 
9789 

9715 
9755 
9792 

9720 

9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9736 
9774 
9810 

9740 
9778 
9813 

2 
2 
2 

3      3 
3      3 
2       3 

79 
80 
81 

9816 
9848 
9877 

9820 

9851 
9880 

9823 
9854 
9882 

9907 
9930 
9949 

9826 

9857 
9885 

9829 
9860 
9888 
9912 
9934 
9952 

9833 
9863 
9890 

9914 
9936 
9954 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I             2 
0             I 
0             1 

2        3 
2        2 
2        2 

82 
83 
84 

9903 
9925 
9945 
9962 

9905 
9928 

9947 
9963 

9910 
9932 
9951 
9966 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

O 
0 
0 

2        2 
I         2 
I         1 

85 

9965 

9968 

9969 
9981 
9990 
9997 

9971 

9972 

9973 

9974 

O     O 

I         I 

86 
87 
88 

9976 
9986 
9994 

9977 
9987 
9995 

9978 
9988 
9995 

9979 
9989 
9996 

9980 
9990 
9996 

9982 
9991 
9997 

9983 
9992 
9997 

9984 
9993 
9998 

9985 
9993 
9998 

0    O 
0     0     O 
000 

I         I 
I         I 
O        O 

89 

9998 

9999 

9999 

9999 

9999 

1.000 
nearly 

1.  000 

nearly 

I.OOO 
icarlv 

I.OOO 
nearly 

i  I.OOO 
nearly 

000 

0        0 

62 


NATURAL  TANGENTS 


0 

6 

12 

18 

24' 

30' 

36 

42 

48' 

54' 

123 

4      5 

0° 

.0000 

0017 

0035 

0209 
0384 
0559 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

0332 
0507 
0682 

369 

12         14- 

1 

2 
3 

•0175 

•0349 
.0524 

0192 
0367 
0542 

0227 
0402 
0577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

0314 
0489 
0664 

369 
369 
369 

12         15 
12          15 
12          I5 

4 
5 
6 

.0699 
.0875 
.1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
"39 

0805 
0981 
"57 

1334 
1512 
1691 

0822 
0998 

"75 

0840 
1016 
1192 

0857 
1033 

1210 

369' 
369 
369 

12          15 
12          IS 
12          I5 

7 
8 
9 

.1228 
.1405 
.1584 

1246 

1423 
1602 

1263 
1441 
1620 

1281 

1459 
1638 

1299 
1477 
1655 

1317 
1495 
1673 

1352 
1530 
1709 

1370 

1548 
1727 

1388 
1566 
1745 

369 
369 
369 

12         15 
12         15 
12         I5 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

3      6     9^ 

12         15 

11 
12 
13 

.1944 
.2126 
.2309 

1962 
2144 
2327 

1980 
2162 
2345 

1998 
2180 
2364 

2016 
2199 

2382 

2035 
2217 
2401 

2053 
2235 
2419 

2071 
2254 
2438 

2089 
2272 
2456 

2107 
2290 
2475 
2661 
2849 
3038 

369 
369 
369 

12         15 
12         15 
12          15 

14 
15 
16 

•2493 
.2679 
.2867 

2512 
2698 
2886 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 

2623 
2811 
3000 

2642 
2830 
3019 

369 
369 
369 

12         l6 

13       16 
J3       J6 

17 
18 
19 

•3057 
3249 
•3443 

3076 
3269 
3463 

3096 
3288 
3482 

3H5 
3307 
3502 

3134 
3327 
3522 

3153 
3346 
3541 

3172 
3365 
356i 

3I91 

3385 
358i 

3211 
3404 
3600 

3230 
3424 
3620 

3      6    10 
3      6    10 
3      6    10 

13       16 
13       16 
13       17 

20 

•3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3      7    '0 

ij       J7 

21 
22 
23 

.3839 
.4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

4307 

3919 
4122 

4327 

3939 
4142 

4348 

3959 
4163 

4369 

3978 
4183 
4390 

4000 
4204 
44" 

4020 
4224 
4431 

3      7    Jo 
3      7    I0 
3'     7    JO 

J3       J7 
J4       J7 
14       17 

24 
25 
26 

•4452 
.4663 
.4877 

4473 
4684 
4899 

4494 
4706 
4921 

45i5 
4727 
4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 

479' 
5008 

4599 
4813 
5029 

4621 
4834 
5051 

4642 
4856 
5073 

4      7    J° 
4      7    ii 
4      7    ii 

14       18 
14       18 
15       18 

27 
28 
29 

•5095 
•5317 

•5543 

5"7 
5340 
5566 

5139 
5362 

5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

5250 
5475 
5704 

5272 
5498 
5727 

5295 
5520 
5750 

4      7    M 
4811 
4      8    12 

15       J» 
J5       '9 
J5      '9 

30 

•5774 

5797 

5820 

5844 

5867 

5890 

5914 

5938 

596i 

5985 

4      8    .2 

16      20 

31 
32 
33 

.6009 
.6249 
.6494 

6032 
6273 
6519 
6771 
7028 
7292 

6056 
6297 
6544 

6080 
6322 
6569 

6lO4 
6346 
6594 

6128 
6371 
6619 

6152 

6395 
6644 

6899 
7159 
7427 

6176 
6420 
6669 
6924 
7186 
7454 

6200 
6445 
6694 

6224 
6469 
6720 

4      8    12 
4812 
4      S    13 

1  6      20 
16      20 

I7          21 

34 
35 
36 

6745 
.7002 
•7265 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 

7373 

6873 
7133 
7400 

6950 
7212 
7481 

6976 

7239 
7508 

4      9    J3 
4      9    J3 
5      9    J4 

17         21 

l8          22 

18       23 

37 
38 
39 

•7536 
•7813 
.8098 

7563 

7841 
8127 

7590 
7869 
8156 

7618 
7898 
8185 

7646 
7926 
8214 

7673 
7954 

8243 

7701 
7983 

8273 

7729 
8012 
8302 

7757 
8040 
8332 

7785 
8069 
8361 

5     9    J4 
5    J°    J4 
5    J°    J5 

18         23 
19         24 
20         =4 

40 

.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

5    io    15 

2O         25 

41 
42 
43 

.8693 
.9004 
•9325 

8724 
9036 
9358 

8754 
9067 
9391 

8785 
9099 

9424 

8816 
9131 
9457 

8847 
9163 
9490 

8878 
9195 
9523 

8910 

9228 
9556 

8941 
9260 
9590 

8972 
9293 
9623 

5    io    16 
5    ii    16 
6    ii    17 

21         26 
SI         27 
22         28 

44 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

99&5 

6    ii    17 

23         29 

NATURAL  TANGENTS. 


63 


0 

6' 

12 

18 

24' 

30 

36 

42' 

48 

54' 

123 

4    5 

45° 

46 
47 
48 

I.OOOO 

0035 

0070 

0105 

0141 

0176 

O2I2 

0575 
095  J 
1343 

0247 

0283 

0319 

6     12      18 

24   30 

1-0355 
1.0724 

1.1106 

0392 
0761 
H45 

0428 
0799 
1184 

0464 
0837 
1224 

0501 

0875 
1263 

0538 
0913 
1303 

0612 
0990 
1383 

0649 
1028 
1423 

0686 
1067 
1463 

6     12     18 

6   13    19 
7    13    20 

25    3' 
25    32 
26    33 

49 
50 
51 

1.1504 
1.1918 
1-2349 

1544 
1960 

2393 

1585 

2002 

2437 

1626 
2045 
2482 

1667 
2088 
2527 

1708 
2131 
2572 

I75C 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 
2753 

7    *4    21 
7    H    22 
8    15    23 

28     34 
29     36 
30     38 

52 
53 
54 

1.2799 
1.3270 
1.3764 

2846 
3319 
3814 

2892 
3367 

3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
3514 
4019 

3079 
3564 
4071 

3127 
3613 
4124 

3175 
3663 
4176 

3222 

3713 
4229 

8    16    23 
8    16    25 
9    »7    *6 

3'     39 
33    4' 
34     43 

10  IVO  OCO 

10  |iO  10  10 

1.4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9    18    27 

36    45 

1.4826 

J-5399 
1.6003 

4882 
5458 
6066 

4938 
5517 
6128 

4994 
5577 
6191 

5051 
5637 
6255 

5108 

5697 
6319 

5166 

5757 
6383 

5224 
5818 
6447 

5282 
5880 
6512 

5340 
5941 
6577 

10    19    29 
io   20   30 

11      21      32 

38    48 
40    50 
43    53 

59 
60 
61 
62 
63 
64 

1.6643 
1.7321 
1.8040 

6709 
7391 

8115 

6775 
7461 

8190 

6842 
7532 
8265 

6909 
7603 
8341 

6977 

7675 
8418 

7045 
7747 
8495 

7"3 

7820 
8573 

7182 

7893 
8650 

7251 
7966 
8728 

Ji    23    34 

12     24     36 
13     26     38 

45    56 
48    60 
5i    64 

1.8807 
1.9626 
2.0503 

8887 
9711 
0594 

8967 

9797 
0886 

9047 

9883 
0778 

9128 
9970 
0872 

9210 

6057 
0965 

3292 
0145 
1060 

9375 
0233 

"55 

9458 
6323 
1251 

9,542 
0413 
1348 

I4     27     41 
»5    29    44 
16    31    47 

S5    68 
53    73 
63    78 

~68~8s 

65 

2.1445 

1543 

1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

'7    34    5» 

66 
67 
68 

2.2460 
2-3559 
24751 

2566 

3673 
4876 

2673 

3789 
5002 

2781 
3906 
5129 

2889 
4023 
5257 

2998 
4142 
5386 

3109 
4262 
5517 

3220 
4383 
5649 

3332 
4504 
5782 

3445 
4627 
59*6 

'8    37    55 

20     40     60 

«    43    65 

74    92 
79    99 
87  108 

69 
70 
71 

2.6051 

2-7475 
2.9042 

6187 
7625 
9208 

6325 
7776 
9375 

6464 
7929 
9544 

6605 
8083 
9714 

6746 
8239 
9887 

6889 
8397 
0061 

7034 
8556 
0237 

7179 
8716 
6415 

7326 

8878 
0595 

24    47    7' 

26    52    78 
29    58    87 

95  "8 
104  130 
115  M4 

72 
73 
74 

3-0777 
32709 

3.4874 

0961 
2914 
5105 

1146 
3122 
5339 

1334 
3332 
5576 

1524 
3544 
5816 

1716 

3759 
6059 

1910 
3977 
6305 

2106 
4197 
6554 

2305 
4420 
'6806 

2506 
4646 
7062 

32    64    96 
36    72  108 

41     82  122 

129  1161 

144  180 
162  203 

75 

3.7321 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

9812 

46    94  139 

186  232 

76 
77 
78 

4.0108 

4-3315 
4.7046 

0408 
3662 
7453 

0713 
4015 
7867 

JO22 

4374 

8288 

1335 
4737 
8716 

1653 
5107 
9152 

1976 
5483 
9594 

2303 
5864 
0045 

2635 
6252 
0504 

2972 
6646 
0970 

53  107  160 
62  124  186 
73146219 

214  267 
248  310 
292  365 

79 
80 
81 

5.1446 
5.6713 

1929 
7297 
3859 

2422 
7894 
4596 

292^ 
8502 
5350 

3435 
9I24 
6122 

3955 
9758 
6912 

4486 
0405 
7920 

5026 
1066 

8548 

5578 
1742 
9395 

6140 

?<n? 

87  175  262 

350  437 

6.313^ 

0264 

82 
83 
84 

7-H54 

8.1443 
9-5144 

2066 
2636 
9.677 

3002 
3863 
9-845 

3962 
5126 
IO.O2 

4947 
6427 

IO.2O 

5958 
7769 
10.39 

6996 
9152 
10.58 

8062 

0579 
10.78 

9158 
2052 
10.99 

0285 
3572 
n.  20 

Difference  -  col- 
umns  cease  to  be 
useful,   owing    to 
the    rapidity  with 
which    the    value 
of  the  tangent 
changes. 

85 
86 
87 
88 

n-43 

n.66 

11.91 

12.16 

1243 

12.71 

13.06 

13-30 

13-62 

1395 

14.30 
19.08 
28.64 

14.67 
19.74 
30.14 

15-06 
20.45 
31-82 

15.46 
21.20 
33.69 

15.89 
22.02 
35-80 

16.35 
22.90 
38.19 

16.83 
23-86 
40.9 

17-34 
24.90 
44.07 

17.89 
26.0; 
47-74 

18.46 
27.27 
52.08 

89 

57-29 

63-66 

71.62 

81.8 

95-49 

114.6 

143-2 

191.0 

286.5 

573.0 

64 


LOGARITHMS. 


0 

1 

2 

3 

4 

•5 

6 

7 

8 

9 

123 

456 

789 

10 

oooo 

0043 

0086 

0128 

0170 

O2I2 

0253 

0294 

0334 

0374 

Use  Ta   le  second 
page  fo!lowin;4 

11 

12 
13 

0414 
0792 
1139 

0453 
0828 

"73 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

6719 
1072 
1399 

0755 
1106 
T430 

4811 
3  7  10 
3  6  10 

15  19  23 

14    17    21 

13  16  19 

26  30  34 
24  28  31 
23  26  29 

14 
15 
IS 
17 
18 
19 
20 
21 
22 
23 

1461 
1761 
2041 

1492 
1790 
2068 

1523 
1818 
2095 

1553 
1847 

2122 

1584 
1875 
2148 

1614 
1903 
2175 

1644 
!93i 

2201 

1673 
1959 
2227 

i?£>3 
1967 
2253 

1732 
2014 

2279 

369 
368 
3  5  8 

12  15  18 

II  14  17 
II  13  16 

21    24    27 
20   22    25 

18  21  24 

2304 

2553 
2788 

2330 
2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 

2878 

2430 
2672 
29OO 

2455 
2695 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

5  7 
5  7 
4  7 

10    12    15 

9  12  J4 
9  "  *3 

17    2O   22 

16  19  21 
16  18  20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

4  6 

8  ii  13 

15  17  19 

3222 
3424 

3617 

3243 
3444 
3636 

3263 
3464 

3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 

37" 

3345 
354i 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

4  6 
4  6 
4  6 

8    10    12 
8    10    12 

7    9  ii 

14  16  18 
14  15  17 
13  '5  »7 

24 
25 
26 

3802 

3979 
4rto 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 

4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

4   5 
3  5 
3   5 

7    9  ii 
7    9  10 
7    8  10 

12  14  16 

12   14   15 
II    13    15 

27 
28 

29 

4314 
4472 
4624 

4330 
H87 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4814 

43.78 

4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 

4728 

4440 
4594 
4742 

4456 
4609 
4757 

3   5 
3  5 
3  4 

6>8    9 
689 
679 

II    13    14 
II    12    14 

10    12    13 

30 
31 
32 
33 

477i 

4786 

4800 

4829 

4843 

4857 

4871 

4886 

4900 

3   4 

679 

10    II    13 

4914 
5051 
5185 

1928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 

5237 

4983 
5"9 
5250 

4997 
5132 
5263 

5011 

5145 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

3  4 
3  4 
3  4 

678 
5    7    8 
5    6    8 

10    II     J2 
9    11    12 

9  10  12 

34 
35 
36 
37 
38 
39 

5315 
544i 
5563 

5328 
5453 

5575 

5340 
5465 
5587 

5353 
5478 
5599 
5717 
5832 
5944 

5366 

549<> 
5611 

5378 
5502 
5623 

5391 

5514 
5635 

5403 

5527 
5647 

5416 

5539 
5658 

5428 

5551 
5670 

3  4 
4 

4 

5    6    8 
5    6    7 
5    6    7 

9  10  ii 
9-io  ii 
8  10  ii 

5682 
5798 
5911 

5694 
S8o9 
5922 

5705 
5821 

5933 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 

5999 

5786 

5899 
6010 

3 
3 

3 

5    6    7 
5    6    7 
457 

8    9  10 
8    9  10 
8    9  10 

40 
41 
42 
43 

6021 
6128 
6232 
6335 

6031 

6042 

605.3 

6064 

6075 

6085 

6096 

6107 

6117 

3 

4    5    6 

8    9  10 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

3 
3 
r       3 

456 
4    5    6 
4    5    6 

789 
7     8    9 
7    8    Q 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
6551 
6646 

6464 
6561 
6656 

6474 

6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
65-99 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

3 
3 
3 

456 
4    5    6 
456 

7     8    9 
7     8    9 
7     7     8 

47 
48 
49 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 
7007 

6749 

6839 
6928 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

3 
3 
3 

455 

445 
445 

6    7    8 
6    7    fe 
6    7    8 

50 
51 
52 
53 

6990 

6998 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

3 

345 

6     7     8 

7076 
7160 
7243 

7084 
7168 
7251 

7093 

7177 
7259 

7101 
7185 
7267 

7348 

7110 
7193 

7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7143 
7226 
7308 

7152 
7235 
7316 

3 

2 
2 

345 
345 
345 

6    7     8 
6    7     7 
667 

54 

7324 

7332 

7340 

7356 

7364 

7372 

738o 

7388 

7396 

I    2    2 

345 

6    6    7 

LOGARITHMS 


65 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

55 

740-1 

7412 

7419 

7427 

7435 

7443 

745J 

7459 

7466 

7474 

I  2  2 

345 

5  6  7 

56 
57 
58 

7482 
7559 
7634 

7490 
7566 
7642 

7497 
7574 
7649 

7505 
7582 
7657 

7513 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

7551 
7627 
770) 

I  2 
I  2 
I  2 

3  4  5 
345 
344 

5  6  7 
5  6  7 
5  6  7 

59 
60 
61 

7709 
7732 

78<n 

7716 
7789 
7860 

7723 
7796 
7868 

773i 
7803 
7S75 

7735 
7810 
7882 

7745 
7818 
7889 

7752 
7825 
7896 

7760 
7832 
7903 

7767 

7839 
7910 

7774 
7846 
79T7 

I  2 
I  2 
I  2 

344 
344 
344 

5  6  7 
5  6  t 
5  6  6 

62 
63 
64 

7924 

7993 
8062 

7931 
Sooo 
8069 

7938 
8007 

8075 

7945 
8014 
8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

798o 
8048 
8116 

7987 
8055 
812? 

I  2 

I  2 
I  2 

334 
334 
334 

5  6  6 

5  5  6 
5  5  ( 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

1  '2 

334 

5  5  <- 

66 
67 
68 

8195 
8261 

8325 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 

8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

I  2 

334 

4  5  6 

69 
70 
71 

8388 
8451 
8513 

8395 
8457 
8519 

8401 
8463 

8525 

8407 
8470 
853i 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 

8555 

8439 
8500 
8561 

8445 
8506 

8567 

1  2 

234 

4  5  6 

I  '  2 

234 

455 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 
8762 

S591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 

8745 

I  2 

I  2 
I  2 

234 
2  3  4 
234 

455 
455 
455 

75 

875i 

8756 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

I  2 

233 

4  5  5 

76 
77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

885-9 
8915 

8971 

I  2 

I  2 

233 
233 

445 
445 

79 
80 
81 

8976 
9031 

9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 
9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 
9180 
9232 
9284 

9025 
9079 
9133 

4  4  5 

445 

82 
83 
84 

9138 
9191 
9243 

9r43 
9196 

9248 

9149 
9201 
9253 

9*54 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 

9279 

9186 
9238 
9289 

I  2 
I  2 

233 
233 

445 
445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 
87 
88 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

936o 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

1  2 

0  1 
O  I 

233 

2     3 

2      3 

445 
3  4  4 
3  4  4 

89 
90 
91 

9494 
9542 
959° 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

95i8 
9566 
9614 
9661 
9708 
9754 

9523 
957' 
9619 

9528 
9576 
9624 

9533 
958i 
9628 

953S 
9586 
9633 

0  I 
0  I 
0  I 

2    3 

2      3 

2    3 

~*~   3 

2    3 
2  •   3 

3  4  4 
344 

3  4  4 

92 
93 
94 

9638 
9685 
973' 

9643 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 

9657 
9703 
9750 

9666 
9713 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

O  1 
O  1 
O  I 

344 
344 
3  4  4 

95 

9777 

9782 

9786 

9-791 

9795 

9800 

9805 

9809 

9814 

9818 

0  J 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 

)92I 

9836 
9881 
9926 

9841 
9886 
9930 

9«45 
9890 
9934 

9850 
9«94 
9939 

9854 
9899 

9943 

9859 
9903 
9948 

9863 
99oS 
9952 

0  I 
0  I 
0  I 

2    3 
2    3 
2    3 

344 

344 
344 

99 

9956 

9961 

9965  9969 

9974 

9978 

9983 

9987 

999  * 

9996 

O  I  1 

223 

334 

66 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

ooooo 

043 

087 

130 

173 

217 

'^60 

303 

346 

389 

101 
102 
103 

432 
860 

01284 

475 
903 
326 

5i8 
945 
368 

561 
c,88 
410 

604 
030 
452 

647 
072 
494 

689 
H5 
536 

732 
157 

578 

775 
199 
620 

817 
242 
662 

104 
105 
106 

703 

02  119 

531 

745 
160 
572 

787 
202 
612 

828 
243 
653 

870 
284 
694 

912 
325 

735 

953 
366 
776 

995 
407 
816 

036 

449 

857 

078 
490 
898 

107 
108 
109 

938 
03342 

743 

979 

383 
782 

019 

423 

822 

060 
463 

862 

100 

503 
902 

141 

543 
941 

181 

583 
981 

222 
623 
021 

262 
663 
060 

302 
703 

IOO 

To  find  the  logarithm  of  a  number:  First,  locate  in  the 
table  the  mantissa  which  lies  in  line  with  the  first  two  figures  of  the 
number  and  underneath  the  third  figure,  then  increase  this  mantissa 
by  an  amount  depending  upon  the  fourth  figure  of  the  number  and 
found  by  means  of  the  interpolation  columns  at  the  right;  secondly, 
determine  the  characteristic,  or  the  exponent  of  that  integer  power 
of  10  which  lies  next  in  value  below  the  number;  for  example, 
log  600;=  0.7782  -(-  2. ;  log  73-46=  0.8661  +  1. ; 

log  ,006=0.7782-3.;  log  .7346=0.8661-1.; 

log -6.003=  Q-.7784  +  0. ;  log  734j9=  0.8662  +  3. 

The  logarithm  of  a  product  of  two  or  more  numbers  is  the  sum  of 
the  logarithms  of  its  factors;  for  example, 

log.  (.0821  X  463.2)=  (0.9143-2.)  +  (0.6658  +  2.)  =  0.5801  +  1. 
The  logarithm  of  a  quotient  is  the  difference  between  the  logar- 
ithms of  the  dividend  and  divisor;  for  example, 

log.  (.5321 -916)  =(0.7260-1.)  -  (0.9619 +  2.)  =  0.7641-4. 
The  logarithm  of  a  power  or  root  of  a  number  is  the  exponent 
times  the  logarithm  of  the  number;  for  example, 

log  N/.863)8=3/2  X  (0.9360— l.)  =  0.9040-l. 

To  find  the  number  from  its  logarithm:  Locate  in  the  table 
the  mantissa  next  less  than  the  given  mantissa,  then  join  the  figure 
standing  above  it  at  the  top  of  the  table  to  the  two  figures  at  the 
extreme  left  on  the  same  line  as  the  mantissa,  and  finally  to  these 
three  join  the  figure  at  the  top  of  the  interpolation  column  which 
contains  the  difference  between  the  two  mantissae.  In  the  four- 
figure  number  thus  found,  so  place  the  decimal  point  that  the 
number  shall  be  the  product  of  some  number,  that  lies  between 
1  and  10,  by  a  power  of  10  whose  exponent  is  the  characteristic 
of  the  logarithm.  For  example, 

antilog  (0.6440 +  3)  =  4405; 
antilog  (0.3069 -2)  =  .02027. 

Caution.  Tn  adding  and  subtracting  logarithms  it  is  well  to 
remember  that  the  mantissa  is  always  essentially  positive  and  may 
or  may  not  therefore,  have  the  same  sign  as  its  characteristic. 


INDEX 


Aberration,  chromatic 

study  of        -  -     34 

spherical  study  of  -     35 

Angles,    measuring,    with 

spectrometer  24 

Astigmatic  pencil  18 

Astigmatism,  in  lens          -     35 
Brewster's  Law  55 

Cardinal  Points  of  Thick 

Lens  37 

Caustic  Curve  -     18 

Chord,  Major  -  -       8 

Convention  of  Signs,  opti- 
cal        -  -     11 
Conjugate  Points      -         -     21 
Deviation,  Angle  of  mini- 
mum   -                             -     28 
Dust  Figures    -  5 
Eyepiece,  Gauss,  use  of  -     25 
focal  length   of                     40 
Focal  Distance,  first  and 
second,  principal,  defined    21 
direct  measurement  of        33 
Formula,   for  mirror,  sin- 
gle   refracting    surface, 
lens       -                             -     12 
Fraunhofer  lines,  the         -     46 
Grating  Constant,  the      -     51 
Harmony                              -       9 
Image,  formed  by  convex 

reflecting  surface  -     15 

Image:  graphic  determin- 
ation of;  using  principal 
planes  37 

Kundt's   methqd    for   de- 
termining   the    velocity 
of  sound  in  solids        -       5 
Lenses,  thin,  defined          -     33 
Magnifying  power  of  tele- 
scope   -  -     40 


Mirrors,  concave,  method 
of  determining  radius  of 

curvature      -  13 

convex                              -  13 

Model  Eye,  study  of  21 

Newton's  Rings  48 

Nicol's  polarizing  prisms  -  52 
Nodal     Points    of     Thick 

Lenses:  Defined    -  37 

Determined  38 

Path  of  any  oblique  ray   -  12 

Polarization,  plane  of        -  55 

angle  of  maximum         -  55 
Principal  Points  of  Thick 

Lenses                              -  36 
Radius   of   curvature,    by 

reflection       -                   -  15 
Refraction  trough  20 
Refraction,     at     a    single 
plane  surface                   -  17 
at  a  single  spherical  sur- 
face                                  -  20 
Refractometer,  Pulfrich's  30 
Scale,  the  natural  8 
Spectra    -                            -  45 
Spectrometer,  adjustments  24 
Spectroscope,  direct  vision  44 
Spherometer,  the  Ring      -  13 
the  Peg                             -  14 
the  Aldis  peg                  -  14 
Telescope,  resolving  power 
of                                       -  40 
magnifying  power  of      -  40 
Thick  Lenses,  defined  36 
Triad,  major    -  8 
Velocity  of  sound,  in  tubes  7 
in  rods                              -  5 
Vibration  of  wires  9 
Young's  Modulus,of  wires, 
from     longitudinal     vi- 
brations        -                   -  9 
of  rods,  from  dust  figures  7 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


NOV  21  1947 


!5Apr52LU 

.      26ANov'52FA 


*       8 


JAN  5     195t 


;  • 

3lan'58WWXi 
REC'D  LD 

DEC    61957 

16Mar'59Tj 


REC'D  LD 

1959 


REC'D 

APR  30  i3o 


REC'D  LD 
SIP  8  0  1962 


LD  21-100m-9,547(A5702sl6)476 


ru    I 


7 


' 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


.» •> 


•Si?    "•'    ;'% 


